diff -r 0c666a0c57d7 -r aef235b965bb thys2/Journal/Paper.thy~
--- a/thys2/Journal/Paper.thy~	Fri Jan 07 22:25:26 2022 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2277 +0,0 @@
-(*<*)
-theory Paper
-imports 
-   "../Lexer"
-   "../Simplifying" 
-   "../Positions"
-
-   "../SizeBound" 
-   "HOL-Library.LaTeXsugar"
-begin
-
-lemma Suc_0_fold:
-  "Suc 0 = 1"
-by simp
-
-
-
-declare [[show_question_marks = false]]
-
-syntax (latex output)
-  "_Collect" :: "pttrn => bool => 'a set"              ("(1{_ \<^latex>\<open>\\mbox{\\boldmath$\\mid$}\<close> _})")
-  "_CollectIn" :: "pttrn => 'a set => bool => 'a set"   ("(1{_ \<in> _ |e _})")
-
-syntax
-  "_Not_Ex" :: "idts \<Rightarrow> bool \<Rightarrow> bool"  ("(3\<nexists>_.a./ _)" [0, 10] 10)
-  "_Not_Ex1" :: "pttrn \<Rightarrow> bool \<Rightarrow> bool"  ("(3\<nexists>!_.a./ _)" [0, 10] 10)
-
-
-abbreviation 
-  "der_syn r c \<equiv> der c r"
-
-abbreviation 
-  "ders_syn r s \<equiv> ders s r"
-
-  abbreviation 
-  "bder_syn r c \<equiv> bder c r"
-
-abbreviation 
-  "bders_syn r s \<equiv> bders r s"
-
-
-abbreviation
-  "nprec v1 v2 \<equiv> \<not>(v1 :\<sqsubset>val v2)"
-
-
-
-
-notation (latex output)
-  If  ("(\<^latex>\<open>\\textrm{\<close>if\<^latex>\<open>}\<close> (_)/ \<^latex>\<open>\\textrm{\<close>then\<^latex>\<open>}\<close> (_)/ \<^latex>\<open>\\textrm{\<close>else\<^latex>\<open>}\<close> (_))" 10) and
-  Cons ("_\<^latex>\<open>\\mbox{$\\,$}\<close>::\<^latex>\<open>\\mbox{$\\,$}\<close>_" [75,73] 73) and  
-
-  ZERO ("\<^bold>0" 81) and 
-  ONE ("\<^bold>1" 81) and 
-  CH ("_" [1000] 80) and
-  ALT ("_ + _" [77,77] 78) and
-  SEQ ("_ \<cdot> _" [77,77] 78) and
-  STAR ("_\<^sup>\<star>" [79] 78) and
-  
-  val.Void ("Empty" 78) and
-  val.Char ("Char _" [1000] 78) and
-  val.Left ("Left _" [79] 78) and
-  val.Right ("Right _" [1000] 78) and
-  val.Seq ("Seq _ _" [79,79] 78) and
-  val.Stars ("Stars _" [79] 78) and
-
-  L ("L'(_')" [10] 78) and
-  LV ("LV _ _" [80,73] 78) and
-  der_syn ("_\\_" [79, 1000] 76) and  
-  ders_syn ("_\\_" [79, 1000] 76) and
-  flat ("|_|" [75] 74) and
-  flats ("|_|" [72] 74) and
-  Sequ ("_ @ _" [78,77] 63) and
-  injval ("inj _ _ _" [79,77,79] 76) and 
-  mkeps ("mkeps _" [79] 76) and 
-  length ("len _" [73] 73) and
-  intlen ("len _" [73] 73) and
-  set ("_" [73] 73) and
- 
-  Prf ("_ : _" [75,75] 75) and
-  Posix ("'(_, _') \<rightarrow> _" [63,75,75] 75) and
- 
-  lexer ("lexer _ _" [78,78] 77) and 
-  F_RIGHT ("F\<^bsub>Right\<^esub> _") and
-  F_LEFT ("F\<^bsub>Left\<^esub> _") and  
-  F_ALT ("F\<^bsub>Alt\<^esub> _ _") and
-  F_SEQ1 ("F\<^bsub>Seq1\<^esub> _ _") and
-  F_SEQ2 ("F\<^bsub>Seq2\<^esub> _ _") and
-  F_SEQ ("F\<^bsub>Seq\<^esub> _ _") and
-  simp_SEQ ("simp\<^bsub>Seq\<^esub> _ _" [1000, 1000] 1) and
-  simp_ALT ("simp\<^bsub>Alt\<^esub> _ _" [1000, 1000] 1) and
-  slexer ("lexer\<^sup>+" 1000) and
-
-  at ("_\<^latex>\<open>\\mbox{$\\downharpoonleft$}\<close>\<^bsub>_\<^esub>") and
-  lex_list ("_ \<prec>\<^bsub>lex\<^esub> _") and
-  PosOrd ("_ \<prec>\<^bsub>_\<^esub> _" [77,77,77] 77) and
-  PosOrd_ex ("_ \<prec> _" [77,77] 77) and
-  PosOrd_ex_eq ("_ \<^latex>\<open>\\mbox{$\\preccurlyeq$}\<close> _" [77,77] 77) and
-  pflat_len ("\<parallel>_\<parallel>\<^bsub>_\<^esub>") and
-  nprec ("_ \<^latex>\<open>\\mbox{$\\not\\prec$}\<close> _" [77,77] 77) and
-
-  bder_syn ("_\<^latex>\<open>\\mbox{$\\bbslash$}\<close>_" [79, 1000] 76) and  
-  bders_syn ("_\<^latex>\<open>\\mbox{$\\bbslash$}\<close>_" [79, 1000] 76) and
-  intern ("_\<^latex>\<open>\\mbox{$^\\uparrow$}\<close>" [900] 80) and
-  erase ("_\<^latex>\<open>\\mbox{$^\\downarrow$}\<close>" [1000] 74) and
-  bnullable ("nullable\<^latex>\<open>\\mbox{$_b$}\<close> _" [1000] 80) and
-  bmkeps ("mkeps\<^latex>\<open>\\mbox{$_b$}\<close> _" [1000] 80) and
-  blexer ("lexer\<^latex>\<open>\\mbox{$_b$}\<close> _ _" [77, 77] 80) and
-  code ("code _" [79] 74) and
-
-  DUMMY ("\<^latex>\<open>\\underline{\\hspace{2mm}}\<close>")
-
-
-definition 
-  "match r s \<equiv> nullable (ders s r)"
-
-
-lemma LV_STAR_ONE_empty: 
-  shows "LV (STAR ONE) [] = {Stars []}"
-by(auto simp add: LV_def elim: Prf.cases intro: Prf.intros)
-
-
-
-(*
-comments not implemented
-
-p9. The condition "not exists s3 s4..." appears often enough (in particular in
-the proof of Lemma 3) to warrant a definition.
-
-*)
-
-
-(*>*)
-
-section\<open>Core of the proof\<close>
-text \<open>
-This paper builds on previous work by Ausaf and Urban using 
-regular expression'd bit-coded derivatives to do lexing that 
-is both fast and satisfies the POSIX specification.
-In their work, a bit-coded algorithm introduced by Sulzmann and Lu
-was formally verified in Isabelle, by a very clever use of
-flex function and retrieve to carefully mimic the way a value is 
-built up by the injection funciton.
-
-In the previous work, Ausaf and Urban established the below equality:
-\begin{lemma}
-@{thm [mode=IfThen] MAIN_decode}
-\end{lemma}
-
-This lemma establishes a link with the lexer without bit-codes.
-
-With it we get the correctness of bit-coded algorithm.
-\begin{lemma}
-@{thm [mode=IfThen] blexer_correctness}
-\end{lemma}
-
-However what is not certain is whether we can add simplification
-to the bit-coded algorithm, without breaking the correct lexing output.
-
-
-The reason that we do need to add a simplification phase
-after each derivative step of  $\textit{blexer}$ is
-because it produces intermediate
-regular expressions that can grow exponentially.
-For example, the regular expression $(a+aa)^*$ after taking
-derivative against just 10 $a$s will have size 8192.
-
-%TODO: add figure for this?
-
-
-Therefore, we insert a simplification phase
-after each derivation step, as defined below:
-\begin{lemma}
-@{thm blexer_simp_def}
-\end{lemma}
-
-The simplification function is given as follows:
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) bsimp.simps(1)[of  "bs" "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bsimp.simps(1)[of  "bs" "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bsimp.simps(2)} & $\dn$ & @{thm (rhs) bsimp.simps(2)}\\
-  @{thm (lhs) bsimp.simps(3)} & $\dn$ & @{thm (rhs) bsimp.simps(3)}\\
-
-\end{tabular}
-\end{center}
-
-And the two helper functions are:
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) bsimp_AALTs.simps(2)[of  "bs\<^sub>1" "r" ]} & $\dn$ & @{thm (rhs) bsimp.simps(1)[of  "bs\<^sub>1" "r" ]}\\
-  @{thm (lhs) bsimp_AALTs.simps(2)} & $\dn$ & @{thm (rhs) bsimp.simps(2)}\\
-  @{thm (lhs) bsimp_AALTs.simps(3)} & $\dn$ & @{thm (rhs) bsimp.simps(3)}\\
-
-\end{tabular}
-\end{center}
-
-
-This might sound trivial in the case of producing a YES/NO answer,
-but once we require a lexing output to be produced (which is required
-in applications like compiler front-end, malicious attack domain extraction, 
-etc.), it is not straightforward if we still extract what is needed according
-to the POSIX standard.
-
-
-
-
-
-By simplification, we mean specifically the following rules:
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm[mode=Axiom] rrewrite.intros(1)[of "bs" "r\<^sub>2"]}\\
-  @{thm[mode=Axiom] rrewrite.intros(2)[of "bs" "r\<^sub>1"]}\\
-  @{thm[mode=Axiom] rrewrite.intros(3)[of  "bs" "bs\<^sub>1" "r\<^sub>1"]}\\
-  @{thm[mode=Rule] rrewrite.intros(4)[of  "r\<^sub>1" "r\<^sub>2" "bs" "r\<^sub>3"]}\\
-  @{thm[mode=Rule] rrewrite.intros(5)[of "r\<^sub>3" "r\<^sub>4" "bs" "r\<^sub>1"]}\\
-  @{thm[mode=Rule] rrewrite.intros(6)[of "r" "r'" "bs" "rs\<^sub>1" "rs\<^sub>2"]}\\
-  @{thm[mode=Axiom] rrewrite.intros(7)[of "bs" "rs\<^sub>a" "rs\<^sub>b"]}\\
-  @{thm[mode=Axiom] rrewrite.intros(8)[of "bs" "rs\<^sub>a" "bs\<^sub>1" "rs\<^sub>1" "rs\<^sub>b"]}\\
-  @{thm[mode=Axiom] rrewrite.intros(9)[of "bs" "bs\<^sub>1" "rs"]}\\
-  @{thm[mode=Axiom] rrewrite.intros(10)[of "bs" ]}\\
-  @{thm[mode=Axiom] rrewrite.intros(11)[of "bs" "r\<^sub>1"]}\\
-  @{thm[mode=Rule] rrewrite.intros(12)[of "a\<^sub>1" "a\<^sub>2" "bs" "rs\<^sub>a" "rs\<^sub>b" "rs\<^sub>c"]}\\
-
-  \end{tabular}
-\end{center}
-
-
-And these can be made compact by the following simplification function:
-
-where the function $\mathit{bsimp_AALTs}$
-
-The core idea of the proof is that two regular expressions,
-if "isomorphic" up to a finite number of rewrite steps, will
-remain "isomorphic" when we take the same sequence of
-derivatives on both of them.
-This can be expressed by the following rewrite relation lemma:
-\begin{lemma}
-@{thm [mode=IfThen] central}
-\end{lemma}
-
-This isomorphic relation implies a property that leads to the 
-correctness result: 
-if two (nullable) regular expressions are "rewritable" in many steps
-from one another, 
-then a call to function $\textit{bmkeps}$ gives the same
-bit-sequence :
-\begin{lemma}
-@{thm [mode=IfThen] rewrites_bmkeps}
-\end{lemma}
-
-Given the same bit-sequence, the decode function
-will give out the same value, which is the output
-of both lexers:
-\begin{lemma}
-@{thm blexer_def}
-\end{lemma}
-
-\begin{lemma}
-@{thm blexer_simp_def}
-\end{lemma}
-
-And that yields the correctness result:
-\begin{lemma}
-@{thm blexersimp_correctness}
-\end{lemma}
-
-The nice thing about the aove
-\<close>
-
-
-
-section \<open>Introduction\<close>
-
-
-text \<open>
-
-
-Brzozowski \cite{Brzozowski1964} introduced the notion of the {\em
-derivative} @{term "der c r"} of a regular expression \<open>r\<close> w.r.t.\
-a character~\<open>c\<close>, and showed that it gave a simple solution to the
-problem of matching a string @{term s} with a regular expression @{term
-r}: if the derivative of @{term r} w.r.t.\ (in succession) all the
-characters of the string matches the empty string, then @{term r}
-matches @{term s} (and {\em vice versa}). The derivative has the
-property (which may almost be regarded as its specification) that, for
-every string @{term s} and regular expression @{term r} and character
-@{term c}, one has @{term "cs \<in> L(r)"} if and only if \mbox{@{term "s \<in> L(der c r)"}}. 
-The beauty of Brzozowski's derivatives is that
-they are neatly expressible in any functional language, and easily
-definable and reasoned about in theorem provers---the definitions just
-consist of inductive datatypes and simple recursive functions. A
-mechanised correctness proof of Brzozowski's matcher in for example HOL4
-has been mentioned by Owens and Slind~\cite{Owens2008}. Another one in
-Isabelle/HOL is part of the work by Krauss and Nipkow \cite{Krauss2011}.
-And another one in Coq is given by Coquand and Siles \cite{Coquand2012}.
-
-If a regular expression matches a string, then in general there is more
-than one way of how the string is matched. There are two commonly used
-disambiguation strategies to generate a unique answer: one is called
-GREEDY matching \cite{Frisch2004} and the other is POSIX
-matching~\cite{POSIX,Kuklewicz,OkuiSuzuki2010,Sulzmann2014,Vansummeren2006}.
-For example consider the string @{term xy} and the regular expression
-\mbox{@{term "STAR (ALT (ALT x y) xy)"}}. Either the string can be
-matched in two `iterations' by the single letter-regular expressions
-@{term x} and @{term y}, or directly in one iteration by @{term xy}. The
-first case corresponds to GREEDY matching, which first matches with the
-left-most symbol and only matches the next symbol in case of a mismatch
-(this is greedy in the sense of preferring instant gratification to
-delayed repletion). The second case is POSIX matching, which prefers the
-longest match.
-
-In the context of lexing, where an input string needs to be split up
-into a sequence of tokens, POSIX is the more natural disambiguation
-strategy for what programmers consider basic syntactic building blocks
-in their programs.  These building blocks are often specified by some
-regular expressions, say \<open>r\<^bsub>key\<^esub>\<close> and \<open>r\<^bsub>id\<^esub>\<close> for recognising keywords and identifiers,
-respectively. There are a few underlying (informal) rules behind
-tokenising a string in a POSIX \cite{POSIX} fashion:
-
-\begin{itemize} 
-\item[$\bullet$] \emph{The Longest Match Rule} (or \emph{``{M}aximal {M}unch {R}ule''}):
-The longest initial substring matched by any regular expression is taken as
-next token.\smallskip
-
-\item[$\bullet$] \emph{Priority Rule:}
-For a particular longest initial substring, the first (leftmost) regular expression
-that can match determines the token.\smallskip
-
-\item[$\bullet$] \emph{Star Rule:} A subexpression repeated by ${}^\star$ shall 
-not match an empty string unless this is the only match for the repetition.\smallskip
-
-\item[$\bullet$] \emph{Empty String Rule:} An empty string shall be considered to 
-be longer than no match at all.
-\end{itemize}
-
-\noindent Consider for example a regular expression \<open>r\<^bsub>key\<^esub>\<close> for recognising keywords such as \<open>if\<close>,
-\<open>then\<close> and so on; and \<open>r\<^bsub>id\<^esub>\<close>
-recognising identifiers (say, a single character followed by
-characters or numbers).  Then we can form the regular expression
-\<open>(r\<^bsub>key\<^esub> + r\<^bsub>id\<^esub>)\<^sup>\<star>\<close>
-and use POSIX matching to tokenise strings, say \<open>iffoo\<close> and
-\<open>if\<close>.  For \<open>iffoo\<close> we obtain by the Longest Match Rule
-a single identifier token, not a keyword followed by an
-identifier. For \<open>if\<close> we obtain by the Priority Rule a keyword
-token, not an identifier token---even if \<open>r\<^bsub>id\<^esub>\<close>
-matches also. By the Star Rule we know \<open>(r\<^bsub>key\<^esub> +
-r\<^bsub>id\<^esub>)\<^sup>\<star>\<close> matches \<open>iffoo\<close>,
-respectively \<open>if\<close>, in exactly one `iteration' of the star. The
-Empty String Rule is for cases where, for example, the regular expression 
-\<open>(a\<^sup>\<star>)\<^sup>\<star>\<close> matches against the
-string \<open>bc\<close>. Then the longest initial matched substring is the
-empty string, which is matched by both the whole regular expression
-and the parenthesised subexpression.
-
-
-One limitation of Brzozowski's matcher is that it only generates a
-YES/NO answer for whether a string is being matched by a regular
-expression.  Sulzmann and Lu~\cite{Sulzmann2014} extended this matcher
-to allow generation not just of a YES/NO answer but of an actual
-matching, called a [lexical] {\em value}. Assuming a regular
-expression matches a string, values encode the information of
-\emph{how} the string is matched by the regular expression---that is,
-which part of the string is matched by which part of the regular
-expression. For this consider again the string \<open>xy\<close> and
-the regular expression \mbox{\<open>(x + (y + xy))\<^sup>\<star>\<close>}
-(this time fully parenthesised). We can view this regular expression
-as tree and if the string \<open>xy\<close> is matched by two Star
-`iterations', then the \<open>x\<close> is matched by the left-most
-alternative in this tree and the \<open>y\<close> by the right-left alternative. This
-suggests to record this matching as
-
-\begin{center}
-@{term "Stars [Left(Char x), Right(Left(Char y))]"}
-\end{center}
-
-\noindent where @{const Stars}, \<open>Left\<close>, \<open>Right\<close> and \<open>Char\<close> are constructors for values. \<open>Stars\<close> records how many
-iterations were used; \<open>Left\<close>, respectively \<open>Right\<close>, which
-alternative is used. This `tree view' leads naturally to the idea that
-regular expressions act as types and values as inhabiting those types
-(see, for example, \cite{HosoyaVouillonPierce2005}).  The value for
-matching \<open>xy\<close> in a single `iteration', i.e.~the POSIX value,
-would look as follows
-
-\begin{center}
-@{term "Stars [Seq (Char x) (Char y)]"}
-\end{center}
-
-\noindent where @{const Stars} has only a single-element list for the
-single iteration and @{const Seq} indicates that @{term xy} is matched 
-by a sequence regular expression.
-
-%, which we will in what follows 
-%write more formally as @{term "SEQ x y"}.
-
-
-Sulzmann and Lu give a simple algorithm to calculate a value that
-appears to be the value associated with POSIX matching.  The challenge
-then is to specify that value, in an algorithm-independent fashion,
-and to show that Sulzmann and Lu's derivative-based algorithm does
-indeed calculate a value that is correct according to the
-specification.  The answer given by Sulzmann and Lu
-\cite{Sulzmann2014} is to define a relation (called an ``order
-relation'') on the set of values of @{term r}, and to show that (once
-a string to be matched is chosen) there is a maximum element and that
-it is computed by their derivative-based algorithm. This proof idea is
-inspired by work of Frisch and Cardelli \cite{Frisch2004} on a GREEDY
-regular expression matching algorithm. However, we were not able to
-establish transitivity and totality for the ``order relation'' by
-Sulzmann and Lu.  There are some inherent problems with their approach
-(of which some of the proofs are not published in
-\cite{Sulzmann2014}); perhaps more importantly, we give in this paper
-a simple inductive (and algorithm-independent) definition of what we
-call being a {\em POSIX value} for a regular expression @{term r} and
-a string @{term s}; we show that the algorithm by Sulzmann and Lu
-computes such a value and that such a value is unique. Our proofs are
-both done by hand and checked in Isabelle/HOL.  The experience of
-doing our proofs has been that this mechanical checking was absolutely
-essential: this subject area has hidden snares. This was also noted by
-Kuklewicz \cite{Kuklewicz} who found that nearly all POSIX matching
-implementations are ``buggy'' \cite[Page 203]{Sulzmann2014} and by
-Grathwohl et al \cite[Page 36]{CrashCourse2014} who wrote:
-
-\begin{quote}
-\it{}``The POSIX strategy is more complicated than the greedy because of 
-the dependence on information about the length of matched strings in the 
-various subexpressions.''
-\end{quote}
-
-
-
-\noindent {\bf Contributions:} We have implemented in Isabelle/HOL the
-derivative-based regular expression matching algorithm of
-Sulzmann and Lu \cite{Sulzmann2014}. We have proved the correctness of this
-algorithm according to our specification of what a POSIX value is (inspired
-by work of Vansummeren \cite{Vansummeren2006}). Sulzmann
-and Lu sketch in \cite{Sulzmann2014} an informal correctness proof: but to
-us it contains unfillable gaps.\footnote{An extended version of
-\cite{Sulzmann2014} is available at the website of its first author; this
-extended version already includes remarks in the appendix that their
-informal proof contains gaps, and possible fixes are not fully worked out.}
-Our specification of a POSIX value consists of a simple inductive definition
-that given a string and a regular expression uniquely determines this value.
-We also show that our definition is equivalent to an ordering 
-of values based on positions by Okui and Suzuki \cite{OkuiSuzuki2010}.
-
-%Derivatives as calculated by Brzozowski's method are usually more complex
-%regular expressions than the initial one; various optimisations are
-%possible. We prove the correctness when simplifications of @{term "ALT ZERO r"}, 
-%@{term "ALT r ZERO"}, @{term "SEQ ONE r"} and @{term "SEQ r ONE"} to
-%@{term r} are applied. 
-
-We extend our results to ??? Bitcoded version??
-
-\<close>
-
-
-
-
-section \<open>Preliminaries\<close>
-
-text \<open>\noindent Strings in Isabelle/HOL are lists of characters with
-the empty string being represented by the empty list, written @{term
-"[]"}, and list-cons being written as @{term "DUMMY # DUMMY"}. Often
-we use the usual bracket notation for lists also for strings; for
-example a string consisting of just a single character @{term c} is
-written @{term "[c]"}. We use the usual definitions for 
-\emph{prefixes} and \emph{strict prefixes} of strings.  By using the
-type @{type char} for characters we have a supply of finitely many
-characters roughly corresponding to the ASCII character set. Regular
-expressions are defined as usual as the elements of the following
-inductive datatype:
-
-  \begin{center}
-  \<open>r :=\<close>
-  @{const "ZERO"} $\mid$
-  @{const "ONE"} $\mid$
-  @{term "CH c"} $\mid$
-  @{term "ALT r\<^sub>1 r\<^sub>2"} $\mid$
-  @{term "SEQ r\<^sub>1 r\<^sub>2"} $\mid$
-  @{term "STAR r"} 
-  \end{center}
-
-  \noindent where @{const ZERO} stands for the regular expression that does
-  not match any string, @{const ONE} for the regular expression that matches
-  only the empty string and @{term c} for matching a character literal. The
-  language of a regular expression is also defined as usual by the
-  recursive function @{term L} with the six clauses:
-
-  \begin{center}
-  \begin{tabular}{l@ {\hspace{4mm}}rcl}
-  \textit{(1)} & @{thm (lhs) L.simps(1)} & $\dn$ & @{thm (rhs) L.simps(1)}\\
-  \textit{(2)} & @{thm (lhs) L.simps(2)} & $\dn$ & @{thm (rhs) L.simps(2)}\\
-  \textit{(3)} & @{thm (lhs) L.simps(3)} & $\dn$ & @{thm (rhs) L.simps(3)}\\
-  \textit{(4)} & @{thm (lhs) L.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & 
-        @{thm (rhs) L.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
-  \textit{(5)} & @{thm (lhs) L.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & 
-        @{thm (rhs) L.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
-  \textit{(6)} & @{thm (lhs) L.simps(6)} & $\dn$ & @{thm (rhs) L.simps(6)}\\
-  \end{tabular}
-  \end{center}
-  
-  \noindent In clause \textit{(4)} we use the operation @{term "DUMMY ;;
-  DUMMY"} for the concatenation of two languages (it is also list-append for
-  strings). We use the star-notation for regular expressions and for
-  languages (in the last clause above). The star for languages is defined
-  inductively by two clauses: \<open>(i)\<close> the empty string being in
-  the star of a language and \<open>(ii)\<close> if @{term "s\<^sub>1"} is in a
-  language and @{term "s\<^sub>2"} in the star of this language, then also @{term
-  "s\<^sub>1 @ s\<^sub>2"} is in the star of this language. It will also be convenient
-  to use the following notion of a \emph{semantic derivative} (or \emph{left
-  quotient}) of a language defined as
-  %
-  \begin{center}
-  @{thm Der_def}\;.
-  \end{center}
- 
-  \noindent
-  For semantic derivatives we have the following equations (for example
-  mechanically proved in \cite{Krauss2011}):
-  %
-  \begin{equation}\label{SemDer}
-  \begin{array}{lcl}
-  @{thm (lhs) Der_null}  & \dn & @{thm (rhs) Der_null}\\
-  @{thm (lhs) Der_empty}  & \dn & @{thm (rhs) Der_empty}\\
-  @{thm (lhs) Der_char}  & \dn & @{thm (rhs) Der_char}\\
-  @{thm (lhs) Der_union}  & \dn & @{thm (rhs) Der_union}\\
-  @{thm (lhs) Der_Sequ}  & \dn & @{thm (rhs) Der_Sequ}\\
-  @{thm (lhs) Der_star}  & \dn & @{thm (rhs) Der_star}
-  \end{array}
-  \end{equation}
-
-
-  \noindent \emph{\Brz's derivatives} of regular expressions
-  \cite{Brzozowski1964} can be easily defined by two recursive functions:
-  the first is from regular expressions to booleans (implementing a test
-  when a regular expression can match the empty string), and the second
-  takes a regular expression and a character to a (derivative) regular
-  expression:
-
-  \begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) nullable.simps(1)} & $\dn$ & @{thm (rhs) nullable.simps(1)}\\
-  @{thm (lhs) nullable.simps(2)} & $\dn$ & @{thm (rhs) nullable.simps(2)}\\
-  @{thm (lhs) nullable.simps(3)} & $\dn$ & @{thm (rhs) nullable.simps(3)}\\
-  @{thm (lhs) nullable.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) nullable.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) nullable.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) nullable.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) nullable.simps(6)} & $\dn$ & @{thm (rhs) nullable.simps(6)}\medskip\\
-
-%  \end{tabular}
-%  \end{center}
-
-%  \begin{center}
-%  \begin{tabular}{lcl}
-
-  @{thm (lhs) der.simps(1)} & $\dn$ & @{thm (rhs) der.simps(1)}\\
-  @{thm (lhs) der.simps(2)} & $\dn$ & @{thm (rhs) der.simps(2)}\\
-  @{thm (lhs) der.simps(3)} & $\dn$ & @{thm (rhs) der.simps(3)}\\
-  @{thm (lhs) der.simps(4)[of c "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) der.simps(4)[of c "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) der.simps(5)[of c "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) der.simps(5)[of c "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) der.simps(6)} & $\dn$ & @{thm (rhs) der.simps(6)}
-  \end{tabular}
-  \end{center}
- 
-  \noindent
-  We may extend this definition to give derivatives w.r.t.~strings:
-
-  \begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) ders.simps(1)} & $\dn$ & @{thm (rhs) ders.simps(1)}\\
-  @{thm (lhs) ders.simps(2)} & $\dn$ & @{thm (rhs) ders.simps(2)}\\
-  \end{tabular}
-  \end{center}
-
-  \noindent Given the equations in \eqref{SemDer}, it is a relatively easy
-  exercise in mechanical reasoning to establish that
-
-  \begin{proposition}\label{derprop}\mbox{}\\ 
-  \begin{tabular}{ll}
-  \textit{(1)} & @{thm (lhs) nullable_correctness} if and only if
-  @{thm (rhs) nullable_correctness}, and \\ 
-  \textit{(2)} & @{thm[mode=IfThen] der_correctness}.
-  \end{tabular}
-  \end{proposition}
-
-  \noindent With this in place it is also very routine to prove that the
-  regular expression matcher defined as
-  %
-  \begin{center}
-  @{thm match_def}
-  \end{center}
-
-  \noindent gives a positive answer if and only if @{term "s \<in> L r"}.
-  Consequently, this regular expression matching algorithm satisfies the
-  usual specification for regular expression matching. While the matcher
-  above calculates a provably correct YES/NO answer for whether a regular
-  expression matches a string or not, the novel idea of Sulzmann and Lu
-  \cite{Sulzmann2014} is to append another phase to this algorithm in order
-  to calculate a [lexical] value. We will explain the details next.
-
-\<close>
-
-section \<open>POSIX Regular Expression Matching\label{posixsec}\<close>
-
-text \<open>
-
-  There have been many previous works that use values for encoding 
-  \emph{how} a regular expression matches a string.
-  The clever idea by Sulzmann and Lu \cite{Sulzmann2014} is to 
-  define a function on values that mirrors (but inverts) the
-  construction of the derivative on regular expressions. \emph{Values}
-  are defined as the inductive datatype
-
-  \begin{center}
-  \<open>v :=\<close>
-  @{const "Void"} $\mid$
-  @{term "val.Char c"} $\mid$
-  @{term "Left v"} $\mid$
-  @{term "Right v"} $\mid$
-  @{term "Seq v\<^sub>1 v\<^sub>2"} $\mid$ 
-  @{term "Stars vs"} 
-  \end{center}  
-
-  \noindent where we use @{term vs} to stand for a list of
-  values. (This is similar to the approach taken by Frisch and
-  Cardelli for GREEDY matching \cite{Frisch2004}, and Sulzmann and Lu
-  for POSIX matching \cite{Sulzmann2014}). The string underlying a
-  value can be calculated by the @{const flat} function, written
-  @{term "flat DUMMY"} and defined as:
-
-  \begin{center}
-  \begin{tabular}[t]{lcl}
-  @{thm (lhs) flat.simps(1)} & $\dn$ & @{thm (rhs) flat.simps(1)}\\
-  @{thm (lhs) flat.simps(2)} & $\dn$ & @{thm (rhs) flat.simps(2)}\\
-  @{thm (lhs) flat.simps(3)} & $\dn$ & @{thm (rhs) flat.simps(3)}\\
-  @{thm (lhs) flat.simps(4)} & $\dn$ & @{thm (rhs) flat.simps(4)}
-  \end{tabular}\hspace{14mm}
-  \begin{tabular}[t]{lcl}
-  @{thm (lhs) flat.simps(5)[of "v\<^sub>1" "v\<^sub>2"]} & $\dn$ & @{thm (rhs) flat.simps(5)[of "v\<^sub>1" "v\<^sub>2"]}\\
-  @{thm (lhs) flat.simps(6)} & $\dn$ & @{thm (rhs) flat.simps(6)}\\
-  @{thm (lhs) flat.simps(7)} & $\dn$ & @{thm (rhs) flat.simps(7)}\\
-  \end{tabular}
-  \end{center}
-
-  \noindent We will sometimes refer to the underlying string of a
-  value as \emph{flattened value}.  We will also overload our notation and 
-  use @{term "flats vs"} for flattening a list of values and concatenating
-  the resulting strings.
-  
-  Sulzmann and Lu define
-  inductively an \emph{inhabitation relation} that associates values to
-  regular expressions. We define this relation as
-  follows:\footnote{Note that the rule for @{term Stars} differs from
-  our earlier paper \cite{AusafDyckhoffUrban2016}. There we used the
-  original definition by Sulzmann and Lu which does not require that
-  the values @{term "v \<in> set vs"} flatten to a non-empty
-  string. The reason for introducing the more restricted version of
-  lexical values is convenience later on when reasoning about an
-  ordering relation for values.}
-
-  \begin{center}
-  \begin{tabular}{c@ {\hspace{12mm}}c}\label{prfintros}
-  \\[-8mm]
-  @{thm[mode=Axiom] Prf.intros(4)} & 
-  @{thm[mode=Axiom] Prf.intros(5)[of "c"]}\\[4mm]
-  @{thm[mode=Rule] Prf.intros(2)[of "v\<^sub>1" "r\<^sub>1" "r\<^sub>2"]} &
-  @{thm[mode=Rule] Prf.intros(3)[of "v\<^sub>2" "r\<^sub>1" "r\<^sub>2"]}\\[4mm]
-  @{thm[mode=Rule] Prf.intros(1)[of "v\<^sub>1" "r\<^sub>1" "v\<^sub>2" "r\<^sub>2"]}  &
-  @{thm[mode=Rule] Prf.intros(6)[of "vs"]}
-  \end{tabular}
-  \end{center}
-
-  \noindent where in the clause for @{const "Stars"} we use the
-  notation @{term "v \<in> set vs"} for indicating that \<open>v\<close> is a
-  member in the list \<open>vs\<close>.  We require in this rule that every
-  value in @{term vs} flattens to a non-empty string. The idea is that
-  @{term "Stars"}-values satisfy the informal Star Rule (see Introduction)
-  where the $^\star$ does not match the empty string unless this is
-  the only match for the repetition.  Note also that no values are
-  associated with the regular expression @{term ZERO}, and that the
-  only value associated with the regular expression @{term ONE} is
-  @{term Void}.  It is routine to establish how values ``inhabiting''
-  a regular expression correspond to the language of a regular
-  expression, namely
-
-  \begin{proposition}\label{inhabs}
-  @{thm L_flat_Prf}
-  \end{proposition}
-
-  \noindent
-  Given a regular expression \<open>r\<close> and a string \<open>s\<close>, we define the 
-  set of all \emph{Lexical Values} inhabited by \<open>r\<close> with the underlying string 
-  being \<open>s\<close>:\footnote{Okui and Suzuki refer to our lexical values 
-  as \emph{canonical values} in \cite{OkuiSuzuki2010}. The notion of \emph{non-problematic
-  values} by Cardelli and Frisch \cite{Frisch2004} is related, but not identical
-  to our lexical values.}
-  
-  \begin{center}
-  @{thm LV_def}
-  \end{center}
-
-  \noindent The main property of @{term "LV r s"} is that it is alway finite.
-
-  \begin{proposition}
-  @{thm LV_finite}
-  \end{proposition}
-
-  \noindent This finiteness property does not hold in general if we
-  remove the side-condition about @{term "flat v \<noteq> []"} in the
-  @{term Stars}-rule above. For example using Sulzmann and Lu's
-  less restrictive definition, @{term "LV (STAR ONE) []"} would contain
-  infinitely many values, but according to our more restricted
-  definition only a single value, namely @{thm LV_STAR_ONE_empty}.
-
-  If a regular expression \<open>r\<close> matches a string \<open>s\<close>, then
-  generally the set @{term "LV r s"} is not just a singleton set.  In
-  case of POSIX matching the problem is to calculate the unique lexical value
-  that satisfies the (informal) POSIX rules from the Introduction.
-  Graphically the POSIX value calculation algorithm by Sulzmann and Lu
-  can be illustrated by the picture in Figure~\ref{Sulz} where the
-  path from the left to the right involving @{term
-  derivatives}/@{const nullable} is the first phase of the algorithm
-  (calculating successive \Brz's derivatives) and @{const
-  mkeps}/\<open>inj\<close>, the path from right to left, the second
-  phase. This picture shows the steps required when a regular
-  expression, say \<open>r\<^sub>1\<close>, matches the string @{term
-  "[a,b,c]"}. We first build the three derivatives (according to
-  @{term a}, @{term b} and @{term c}). We then use @{const nullable}
-  to find out whether the resulting derivative regular expression
-  @{term "r\<^sub>4"} can match the empty string. If yes, we call the
-  function @{const mkeps} that produces a value @{term "v\<^sub>4"}
-  for how @{term "r\<^sub>4"} can match the empty string (taking into
-  account the POSIX constraints in case there are several ways). This
-  function is defined by the clauses:
-
-\begin{figure}[t]
-\begin{center}
-\begin{tikzpicture}[scale=2,node distance=1.3cm,
-                    every node/.style={minimum size=6mm}]
-\node (r1)  {@{term "r\<^sub>1"}};
-\node (r2) [right=of r1]{@{term "r\<^sub>2"}};
-\draw[->,line width=1mm](r1)--(r2) node[above,midway] {@{term "der a DUMMY"}};
-\node (r3) [right=of r2]{@{term "r\<^sub>3"}};
-\draw[->,line width=1mm](r2)--(r3) node[above,midway] {@{term "der b DUMMY"}};
-\node (r4) [right=of r3]{@{term "r\<^sub>4"}};
-\draw[->,line width=1mm](r3)--(r4) node[above,midway] {@{term "der c DUMMY"}};
-\draw (r4) node[anchor=west] {\;\raisebox{3mm}{@{term nullable}}};
-\node (v4) [below=of r4]{@{term "v\<^sub>4"}};
-\draw[->,line width=1mm](r4) -- (v4);
-\node (v3) [left=of v4] {@{term "v\<^sub>3"}};
-\draw[->,line width=1mm](v4)--(v3) node[below,midway] {\<open>inj r\<^sub>3 c\<close>};
-\node (v2) [left=of v3]{@{term "v\<^sub>2"}};
-\draw[->,line width=1mm](v3)--(v2) node[below,midway] {\<open>inj r\<^sub>2 b\<close>};
-\node (v1) [left=of v2] {@{term "v\<^sub>1"}};
-\draw[->,line width=1mm](v2)--(v1) node[below,midway] {\<open>inj r\<^sub>1 a\<close>};
-\draw (r4) node[anchor=north west] {\;\raisebox{-8mm}{@{term "mkeps"}}};
-\end{tikzpicture}
-\end{center}
-\mbox{}\\[-13mm]
-
-\caption{The two phases of the algorithm by Sulzmann \& Lu \cite{Sulzmann2014},
-matching the string @{term "[a,b,c]"}. The first phase (the arrows from 
-left to right) is \Brz's matcher building successive derivatives. If the 
-last regular expression is @{term nullable}, then the functions of the 
-second phase are called (the top-down and right-to-left arrows): first 
-@{term mkeps} calculates a value @{term "v\<^sub>4"} witnessing
-how the empty string has been recognised by @{term "r\<^sub>4"}. After
-that the function @{term inj} ``injects back'' the characters of the string into
-the values.
-\label{Sulz}}
-\end{figure} 
-
-  \begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) mkeps.simps(1)} & $\dn$ & @{thm (rhs) mkeps.simps(1)}\\
-  @{thm (lhs) mkeps.simps(2)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) mkeps.simps(2)[of "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) mkeps.simps(3)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) mkeps.simps(3)[of "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) mkeps.simps(4)} & $\dn$ & @{thm (rhs) mkeps.simps(4)}\\
-  \end{tabular}
-  \end{center}
-
-  \noindent Note that this function needs only to be partially defined,
-  namely only for regular expressions that are nullable. In case @{const
-  nullable} fails, the string @{term "[a,b,c]"} cannot be matched by @{term
-  "r\<^sub>1"} and the null value @{term "None"} is returned. Note also how this function
-  makes some subtle choices leading to a POSIX value: for example if an
-  alternative regular expression, say @{term "ALT r\<^sub>1 r\<^sub>2"}, can
-  match the empty string and furthermore @{term "r\<^sub>1"} can match the
-  empty string, then we return a \<open>Left\<close>-value. The \<open>Right\<close>-value will only be returned if @{term "r\<^sub>1"} cannot match the empty
-  string.
-
-  The most interesting idea from Sulzmann and Lu \cite{Sulzmann2014} is
-  the construction of a value for how @{term "r\<^sub>1"} can match the
-  string @{term "[a,b,c]"} from the value how the last derivative, @{term
-  "r\<^sub>4"} in Fig.~\ref{Sulz}, can match the empty string. Sulzmann and
-  Lu achieve this by stepwise ``injecting back'' the characters into the
-  values thus inverting the operation of building derivatives, but on the level
-  of values. The corresponding function, called @{term inj}, takes three
-  arguments, a regular expression, a character and a value. For example in
-  the first (or right-most) @{term inj}-step in Fig.~\ref{Sulz} the regular
-  expression @{term "r\<^sub>3"}, the character @{term c} from the last
-  derivative step and @{term "v\<^sub>4"}, which is the value corresponding
-  to the derivative regular expression @{term "r\<^sub>4"}. The result is
-  the new value @{term "v\<^sub>3"}. The final result of the algorithm is
-  the value @{term "v\<^sub>1"}. The @{term inj} function is defined by recursion on regular
-  expressions and by analysing the shape of values (corresponding to 
-  the derivative regular expressions).
-  %
-  \begin{center}
-  \begin{tabular}{l@ {\hspace{5mm}}lcl}
-  \textit{(1)} & @{thm (lhs) injval.simps(1)} & $\dn$ & @{thm (rhs) injval.simps(1)}\\
-  \textit{(2)} & @{thm (lhs) injval.simps(2)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1"]} & $\dn$ & 
-      @{thm (rhs) injval.simps(2)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1"]}\\
-  \textit{(3)} & @{thm (lhs) injval.simps(3)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>2"]} & $\dn$ & 
-      @{thm (rhs) injval.simps(3)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>2"]}\\
-  \textit{(4)} & @{thm (lhs) injval.simps(4)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1" "v\<^sub>2"]} & $\dn$ 
-      & @{thm (rhs) injval.simps(4)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1" "v\<^sub>2"]}\\
-  \textit{(5)} & @{thm (lhs) injval.simps(5)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1" "v\<^sub>2"]} & $\dn$ 
-      & @{thm (rhs) injval.simps(5)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>1" "v\<^sub>2"]}\\
-  \textit{(6)} & @{thm (lhs) injval.simps(6)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>2"]} & $\dn$ 
-      & @{thm (rhs) injval.simps(6)[of "r\<^sub>1" "r\<^sub>2" "c" "v\<^sub>2"]}\\
-  \textit{(7)} & @{thm (lhs) injval.simps(7)[of "r" "c" "v" "vs"]} & $\dn$ 
-      & @{thm (rhs) injval.simps(7)[of "r" "c" "v" "vs"]}\\
-  \end{tabular}
-  \end{center}
-
-  \noindent To better understand what is going on in this definition it
-  might be instructive to look first at the three sequence cases (clauses
-  \textit{(4)} -- \textit{(6)}). In each case we need to construct an ``injected value'' for
-  @{term "SEQ r\<^sub>1 r\<^sub>2"}. This must be a value of the form @{term
-  "Seq DUMMY DUMMY"}\,. Recall the clause of the \<open>derivative\<close>-function
-  for sequence regular expressions:
-
-  \begin{center}
-  @{thm (lhs) der.simps(5)[of c "r\<^sub>1" "r\<^sub>2"]} $\dn$ @{thm (rhs) der.simps(5)[of c "r\<^sub>1" "r\<^sub>2"]}
-  \end{center}
-
-  \noindent Consider first the \<open>else\<close>-branch where the derivative is @{term
-  "SEQ (der c r\<^sub>1) r\<^sub>2"}. The corresponding value must therefore
-  be of the form @{term "Seq v\<^sub>1 v\<^sub>2"}, which matches the left-hand
-  side in clause~\textit{(4)} of @{term inj}. In the \<open>if\<close>-branch the derivative is an
-  alternative, namely @{term "ALT (SEQ (der c r\<^sub>1) r\<^sub>2) (der c
-  r\<^sub>2)"}. This means we either have to consider a \<open>Left\<close>- or
-  \<open>Right\<close>-value. In case of the \<open>Left\<close>-value we know further it
-  must be a value for a sequence regular expression. Therefore the pattern
-  we match in the clause \textit{(5)} is @{term "Left (Seq v\<^sub>1 v\<^sub>2)"},
-  while in \textit{(6)} it is just @{term "Right v\<^sub>2"}. One more interesting
-  point is in the right-hand side of clause \textit{(6)}: since in this case the
-  regular expression \<open>r\<^sub>1\<close> does not ``contribute'' to
-  matching the string, that means it only matches the empty string, we need to
-  call @{const mkeps} in order to construct a value for how @{term "r\<^sub>1"}
-  can match this empty string. A similar argument applies for why we can
-  expect in the left-hand side of clause \textit{(7)} that the value is of the form
-  @{term "Seq v (Stars vs)"}---the derivative of a star is @{term "SEQ (der c r)
-  (STAR r)"}. Finally, the reason for why we can ignore the second argument
-  in clause \textit{(1)} of @{term inj} is that it will only ever be called in cases
-  where @{term "c=d"}, but the usual linearity restrictions in patterns do
-  not allow us to build this constraint explicitly into our function
-  definition.\footnote{Sulzmann and Lu state this clause as @{thm (lhs)
-  injval.simps(1)[of "c" "c"]} $\dn$ @{thm (rhs) injval.simps(1)[of "c"]},
-  but our deviation is harmless.}
-
-  The idea of the @{term inj}-function to ``inject'' a character, say
-  @{term c}, into a value can be made precise by the first part of the
-  following lemma, which shows that the underlying string of an injected
-  value has a prepended character @{term c}; the second part shows that
-  the underlying string of an @{const mkeps}-value is always the empty
-  string (given the regular expression is nullable since otherwise
-  \<open>mkeps\<close> might not be defined).
-
-  \begin{lemma}\mbox{}\smallskip\\\label{Prf_injval_flat}
-  \begin{tabular}{ll}
-  (1) & @{thm[mode=IfThen] Prf_injval_flat}\\
-  (2) & @{thm[mode=IfThen] mkeps_flat}
-  \end{tabular}
-  \end{lemma}
-
-  \begin{proof}
-  Both properties are by routine inductions: the first one can, for example,
-  be proved by induction over the definition of @{term derivatives}; the second by
-  an induction on @{term r}. There are no interesting cases.\qed
-  \end{proof}
-
-  Having defined the @{const mkeps} and \<open>inj\<close> function we can extend
-  \Brz's matcher so that a value is constructed (assuming the
-  regular expression matches the string). The clauses of the Sulzmann and Lu lexer are
-
-  \begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) lexer.simps(1)} & $\dn$ & @{thm (rhs) lexer.simps(1)}\\
-  @{thm (lhs) lexer.simps(2)} & $\dn$ & \<open>case\<close> @{term "lexer (der c r) s"} \<open>of\<close>\\
-                     & & \phantom{$|$} @{term "None"}  \<open>\<Rightarrow>\<close> @{term None}\\
-                     & & $|$ @{term "Some v"} \<open>\<Rightarrow>\<close> @{term "Some (injval r c v)"}                          
-  \end{tabular}
-  \end{center}
-
-  \noindent If the regular expression does not match the string, @{const None} is
-  returned. If the regular expression \emph{does}
-  match the string, then @{const Some} value is returned. One important
-  virtue of this algorithm is that it can be implemented with ease in any
-  functional programming language and also in Isabelle/HOL. In the remaining
-  part of this section we prove that this algorithm is correct.
-
-  The well-known idea of POSIX matching is informally defined by some
-  rules such as the Longest Match and Priority Rules (see
-  Introduction); as correctly argued in \cite{Sulzmann2014}, this
-  needs formal specification. Sulzmann and Lu define an ``ordering
-  relation'' between values and argue that there is a maximum value,
-  as given by the derivative-based algorithm.  In contrast, we shall
-  introduce a simple inductive definition that specifies directly what
-  a \emph{POSIX value} is, incorporating the POSIX-specific choices
-  into the side-conditions of our rules. Our definition is inspired by
-  the matching relation given by Vansummeren~\cite{Vansummeren2006}. 
-  The relation we define is ternary and
-  written as \mbox{@{term "s \<in> r \<rightarrow> v"}}, relating
-  strings, regular expressions and values; the inductive rules are given in 
-  Figure~\ref{POSIXrules}.
-  We can prove that given a string @{term s} and regular expression @{term
-   r}, the POSIX value @{term v} is uniquely determined by @{term "s \<in> r \<rightarrow> v"}.
-
-  %
-  \begin{figure}[t]
-  \begin{center}
-  \begin{tabular}{c}
-  @{thm[mode=Axiom] Posix.intros(1)}\<open>P\<close>@{term "ONE"} \qquad
-  @{thm[mode=Axiom] Posix.intros(2)}\<open>P\<close>@{term "c"}\medskip\\
-  @{thm[mode=Rule] Posix.intros(3)[of "s" "r\<^sub>1" "v" "r\<^sub>2"]}\<open>P+L\<close>\qquad
-  @{thm[mode=Rule] Posix.intros(4)[of "s" "r\<^sub>2" "v" "r\<^sub>1"]}\<open>P+R\<close>\medskip\\
-  $\mprset{flushleft}
-   \inferrule
-   {@{thm (prem 1) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]} \qquad
-    @{thm (prem 2) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]} \\\\
-    @{thm (prem 3) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]}}
-   {@{thm (concl) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]}}$\<open>PS\<close>\\
-  @{thm[mode=Axiom] Posix.intros(7)}\<open>P[]\<close>\medskip\\
-  $\mprset{flushleft}
-   \inferrule
-   {@{thm (prem 1) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \qquad
-    @{thm (prem 2) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \qquad
-    @{thm (prem 3) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \\\\
-    @{thm (prem 4) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}
-   {@{thm (concl) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}$\<open>P\<star>\<close>
-  \end{tabular}
-  \end{center}
-  \caption{Our inductive definition of POSIX values.}\label{POSIXrules}
-  \end{figure}
-
-   
-
-  \begin{theorem}\mbox{}\smallskip\\\label{posixdeterm}
-  \begin{tabular}{ll}
-  (1) & If @{thm (prem 1) Posix1(1)} then @{thm (concl)
-  Posix1(1)} and @{thm (concl) Posix1(2)}.\\
-  (2) & @{thm[mode=IfThen] Posix_determ(1)[of _ _ "v" "v'"]}
-  \end{tabular}
-  \end{theorem}
-
-  \begin{proof} Both by induction on the definition of @{term "s \<in> r \<rightarrow> v"}. 
-  The second parts follows by a case analysis of @{term "s \<in> r \<rightarrow> v'"} and
-  the first part.\qed
-  \end{proof}
-
-  \noindent
-  We claim that our @{term "s \<in> r \<rightarrow> v"} relation captures the idea behind the four
-  informal POSIX rules shown in the Introduction: Consider for example the
-  rules \<open>P+L\<close> and \<open>P+R\<close> where the POSIX value for a string
-  and an alternative regular expression, that is @{term "(s, ALT r\<^sub>1 r\<^sub>2)"},
-  is specified---it is always a \<open>Left\<close>-value, \emph{except} when the
-  string to be matched is not in the language of @{term "r\<^sub>1"}; only then it
-  is a \<open>Right\<close>-value (see the side-condition in \<open>P+R\<close>).
-  Interesting is also the rule for sequence regular expressions (\<open>PS\<close>). The first two premises state that @{term "v\<^sub>1"} and @{term "v\<^sub>2"}
-  are the POSIX values for @{term "(s\<^sub>1, r\<^sub>1)"} and @{term "(s\<^sub>2, r\<^sub>2)"}
-  respectively. Consider now the third premise and note that the POSIX value
-  of this rule should match the string \mbox{@{term "s\<^sub>1 @ s\<^sub>2"}}. According to the
-  Longest Match Rule, we want that the @{term "s\<^sub>1"} is the longest initial
-  split of \mbox{@{term "s\<^sub>1 @ s\<^sub>2"}} such that @{term "s\<^sub>2"} is still recognised
-  by @{term "r\<^sub>2"}. Let us assume, contrary to the third premise, that there
-  \emph{exist} an @{term "s\<^sub>3"} and @{term "s\<^sub>4"} such that @{term "s\<^sub>2"}
-  can be split up into a non-empty string @{term "s\<^sub>3"} and a possibly empty
-  string @{term "s\<^sub>4"}. Moreover the longer string @{term "s\<^sub>1 @ s\<^sub>3"} can be
-  matched by \<open>r\<^sub>1\<close> and the shorter @{term "s\<^sub>4"} can still be
-  matched by @{term "r\<^sub>2"}. In this case @{term "s\<^sub>1"} would \emph{not} be the
-  longest initial split of \mbox{@{term "s\<^sub>1 @ s\<^sub>2"}} and therefore @{term "Seq v\<^sub>1
-  v\<^sub>2"} cannot be a POSIX value for @{term "(s\<^sub>1 @ s\<^sub>2, SEQ r\<^sub>1 r\<^sub>2)"}. 
-  The main point is that our side-condition ensures the Longest 
-  Match Rule is satisfied.
-
-  A similar condition is imposed on the POSIX value in the \<open>P\<star>\<close>-rule. Also there we want that @{term "s\<^sub>1"} is the longest initial
-  split of @{term "s\<^sub>1 @ s\<^sub>2"} and furthermore the corresponding value
-  @{term v} cannot be flattened to the empty string. In effect, we require
-  that in each ``iteration'' of the star, some non-empty substring needs to
-  be ``chipped'' away; only in case of the empty string we accept @{term
-  "Stars []"} as the POSIX value. Indeed we can show that our POSIX values
-  are lexical values which exclude those \<open>Stars\<close> that contain subvalues 
-  that flatten to the empty string.
-
-  \begin{lemma}\label{LVposix}
-  @{thm [mode=IfThen] Posix_LV}
-  \end{lemma}
-
-  \begin{proof}
-  By routine induction on @{thm (prem 1) Posix_LV}.\qed 
-  \end{proof}
-
-  \noindent
-  Next is the lemma that shows the function @{term "mkeps"} calculates
-  the POSIX value for the empty string and a nullable regular expression.
-
-  \begin{lemma}\label{lemmkeps}
-  @{thm[mode=IfThen] Posix_mkeps}
-  \end{lemma}
-
-  \begin{proof}
-  By routine induction on @{term r}.\qed 
-  \end{proof}
-
-  \noindent
-  The central lemma for our POSIX relation is that the \<open>inj\<close>-function
-  preserves POSIX values.
-
-  \begin{lemma}\label{Posix2}
-  @{thm[mode=IfThen] Posix_injval}
-  \end{lemma}
-
-  \begin{proof}
-  By induction on \<open>r\<close>. We explain two cases.
-
-  \begin{itemize}
-  \item[$\bullet$] Case @{term "r = ALT r\<^sub>1 r\<^sub>2"}. There are
-  two subcases, namely \<open>(a)\<close> \mbox{@{term "v = Left v'"}} and @{term
-  "s \<in> der c r\<^sub>1 \<rightarrow> v'"}; and \<open>(b)\<close> @{term "v = Right v'"}, @{term
-  "s \<notin> L (der c r\<^sub>1)"} and @{term "s \<in> der c r\<^sub>2 \<rightarrow> v'"}. In \<open>(a)\<close> we
-  know @{term "s \<in> der c r\<^sub>1 \<rightarrow> v'"}, from which we can infer @{term "(c # s)
-  \<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v'"} by induction hypothesis and hence @{term "(c #
-  s) \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> injval (ALT r\<^sub>1 r\<^sub>2) c (Left v')"} as needed. Similarly
-  in subcase \<open>(b)\<close> where, however, in addition we have to use
-  Proposition~\ref{derprop}(2) in order to infer @{term "c # s \<notin> L r\<^sub>1"} from @{term
-  "s \<notin> L (der c r\<^sub>1)"}.\smallskip
-
-  \item[$\bullet$] Case @{term "r = SEQ r\<^sub>1 r\<^sub>2"}. There are three subcases:
-  
-  \begin{quote}
-  \begin{description}
-  \item[\<open>(a)\<close>] @{term "v = Left (Seq v\<^sub>1 v\<^sub>2)"} and @{term "nullable r\<^sub>1"} 
-  \item[\<open>(b)\<close>] @{term "v = Right v\<^sub>1"} and @{term "nullable r\<^sub>1"} 
-  \item[\<open>(c)\<close>] @{term "v = Seq v\<^sub>1 v\<^sub>2"} and @{term "\<not> nullable r\<^sub>1"} 
-  \end{description}
-  \end{quote}
-
-  \noindent For \<open>(a)\<close> we know @{term "s\<^sub>1 \<in> der c r\<^sub>1 \<rightarrow> v\<^sub>1"} and
-  @{term "s\<^sub>2 \<in> r\<^sub>2 \<rightarrow> v\<^sub>2"} as well as
-  %
-  \[@{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s\<^sub>2 \<and> s\<^sub>1 @ s\<^sub>3 \<in> L (der c r\<^sub>1) \<and> s\<^sub>4 \<in> L r\<^sub>2)"}\]
-
-  \noindent From the latter we can infer by Proposition~\ref{derprop}(2):
-  %
-  \[@{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s\<^sub>2 \<and> (c # s\<^sub>1) @ s\<^sub>3 \<in> L r\<^sub>1 \<and> s\<^sub>4 \<in> L r\<^sub>2)"}\]
-
-  \noindent We can use the induction hypothesis for \<open>r\<^sub>1\<close> to obtain
-  @{term "(c # s\<^sub>1) \<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v\<^sub>1"}. Putting this all together allows us to infer
-  @{term "((c # s\<^sub>1) @ s\<^sub>2) \<in> SEQ r\<^sub>1 r\<^sub>2 \<rightarrow> Seq (injval r\<^sub>1 c v\<^sub>1) v\<^sub>2"}. The case \<open>(c)\<close>
-  is similar.
-
-  For \<open>(b)\<close> we know @{term "s \<in> der c r\<^sub>2 \<rightarrow> v\<^sub>1"} and 
-  @{term "s\<^sub>1 @ s\<^sub>2 \<notin> L (SEQ (der c r\<^sub>1) r\<^sub>2)"}. From the former
-  we have @{term "(c # s) \<in> r\<^sub>2 \<rightarrow> (injval r\<^sub>2 c v\<^sub>1)"} by induction hypothesis
-  for @{term "r\<^sub>2"}. From the latter we can infer
-  %
-  \[@{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = c # s \<and> s\<^sub>3 \<in> L r\<^sub>1 \<and> s\<^sub>4 \<in> L r\<^sub>2)"}\]
-
-  \noindent By Lemma~\ref{lemmkeps} we know @{term "[] \<in> r\<^sub>1 \<rightarrow> (mkeps r\<^sub>1)"}
-  holds. Putting this all together, we can conclude with @{term "(c #
-  s) \<in> SEQ r\<^sub>1 r\<^sub>2 \<rightarrow> Seq (mkeps r\<^sub>1) (injval r\<^sub>2 c v\<^sub>1)"}, as required.
-
-  Finally suppose @{term "r = STAR r\<^sub>1"}. This case is very similar to the
-  sequence case, except that we need to also ensure that @{term "flat (injval r\<^sub>1
-  c v\<^sub>1) \<noteq> []"}. This follows from @{term "(c # s\<^sub>1)
-  \<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v\<^sub>1"}  (which in turn follows from @{term "s\<^sub>1 \<in> der c
-  r\<^sub>1 \<rightarrow> v\<^sub>1"} and the induction hypothesis).\qed
-  \end{itemize}
-  \end{proof}
-
-  \noindent
-  With Lemma~\ref{Posix2} in place, it is completely routine to establish
-  that the Sulzmann and Lu lexer satisfies our specification (returning
-  the null value @{term "None"} iff the string is not in the language of the regular expression,
-  and returning a unique POSIX value iff the string \emph{is} in the language):
-
-  \begin{theorem}\mbox{}\smallskip\\\label{lexercorrect}
-  \begin{tabular}{ll}
-  (1) & @{thm (lhs) lexer_correct_None} if and only if @{thm (rhs) lexer_correct_None}\\
-  (2) & @{thm (lhs) lexer_correct_Some} if and only if @{thm (rhs) lexer_correct_Some}\\
-  \end{tabular}
-  \end{theorem}
-
-  \begin{proof}
-  By induction on @{term s} using Lemma~\ref{lemmkeps} and \ref{Posix2}.\qed  
-  \end{proof}
-
-  \noindent In \textit{(2)} we further know by Theorem~\ref{posixdeterm} that the
-  value returned by the lexer must be unique.   A simple corollary 
-  of our two theorems is:
-
-  \begin{corollary}\mbox{}\smallskip\\\label{lexercorrectcor}
-  \begin{tabular}{ll}
-  (1) & @{thm (lhs) lexer_correctness(2)} if and only if @{thm (rhs) lexer_correctness(2)}\\ 
-  (2) & @{thm (lhs) lexer_correctness(1)} if and only if @{thm (rhs) lexer_correctness(1)}\\
-  \end{tabular}
-  \end{corollary}
-
-  \noindent This concludes our correctness proof. Note that we have
-  not changed the algorithm of Sulzmann and Lu,\footnote{All
-  deviations we introduced are harmless.} but introduced our own
-  specification for what a correct result---a POSIX value---should be.
-  In the next section we show that our specification coincides with
-  another one given by Okui and Suzuki using a different technique.
-
-\<close>
-
-section \<open>Ordering of Values according to Okui and Suzuki\<close>
-
-text \<open>
-  
-  While in the previous section we have defined POSIX values directly
-  in terms of a ternary relation (see inference rules in Figure~\ref{POSIXrules}),
-  Sulzmann and Lu took a different approach in \cite{Sulzmann2014}:
-  they introduced an ordering for values and identified POSIX values
-  as the maximal elements.  An extended version of \cite{Sulzmann2014}
-  is available at the website of its first author; this includes more
-  details of their proofs, but which are evidently not in final form
-  yet. Unfortunately, we were not able to verify claims that their
-  ordering has properties such as being transitive or having maximal
-  elements. 
- 
-  Okui and Suzuki \cite{OkuiSuzuki2010,OkuiSuzukiTech} described
-  another ordering of values, which they use to establish the
-  correctness of their automata-based algorithm for POSIX matching.
-  Their ordering resembles some aspects of the one given by Sulzmann
-  and Lu, but overall is quite different. To begin with, Okui and
-  Suzuki identify POSIX values as minimal, rather than maximal,
-  elements in their ordering. A more substantial difference is that
-  the ordering by Okui and Suzuki uses \emph{positions} in order to
-  identify and compare subvalues. Positions are lists of natural
-  numbers. This allows them to quite naturally formalise the Longest
-  Match and Priority rules of the informal POSIX standard.  Consider
-  for example the value @{term v}
-
-  \begin{center}
-  @{term "v == Stars [Seq (Char x) (Char y), Char z]"}
-  \end{center}
-
-  \noindent
-  At position \<open>[0,1]\<close> of this value is the
-  subvalue \<open>Char y\<close> and at position \<open>[1]\<close> the
-  subvalue @{term "Char z"}.  At the `root' position, or empty list
-  @{term "[]"}, is the whole value @{term v}. Positions such as \<open>[0,1,0]\<close> or \<open>[2]\<close> are outside of \<open>v\<close>. If it exists, the subvalue of @{term v} at a position \<open>p\<close>, written @{term "at v p"}, can be recursively defined by
-  
-  \begin{center}
-  \begin{tabular}{r@ {\hspace{0mm}}lcl}
-  @{term v} &  \<open>\<downharpoonleft>\<^bsub>[]\<^esub>\<close> & \<open>\<equiv>\<close>& @{thm (rhs) at.simps(1)}\\
-  @{term "Left v"} & \<open>\<downharpoonleft>\<^bsub>0::ps\<^esub>\<close> & \<open>\<equiv>\<close>& @{thm (rhs) at.simps(2)}\\
-  @{term "Right v"} & \<open>\<downharpoonleft>\<^bsub>1::ps\<^esub>\<close> & \<open>\<equiv>\<close> & 
-  @{thm (rhs) at.simps(3)[simplified Suc_0_fold]}\\
-  @{term "Seq v\<^sub>1 v\<^sub>2"} & \<open>\<downharpoonleft>\<^bsub>0::ps\<^esub>\<close> & \<open>\<equiv>\<close> & 
-  @{thm (rhs) at.simps(4)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \\
-  @{term "Seq v\<^sub>1 v\<^sub>2"} & \<open>\<downharpoonleft>\<^bsub>1::ps\<^esub>\<close>
-  & \<open>\<equiv>\<close> & 
-  @{thm (rhs) at.simps(5)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2", simplified Suc_0_fold]} \\
-  @{term "Stars vs"} & \<open>\<downharpoonleft>\<^bsub>n::ps\<^esub>\<close> & \<open>\<equiv>\<close>& @{thm (rhs) at.simps(6)}\\
-  \end{tabular} 
-  \end{center}
-
-  \noindent In the last clause we use Isabelle's notation @{term "vs ! n"} for the
-  \<open>n\<close>th element in a list.  The set of positions inside a value \<open>v\<close>,
-  written @{term "Pos v"}, is given by 
-
-  \begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) Pos.simps(1)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(1)}\\
-  @{thm (lhs) Pos.simps(2)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(2)}\\
-  @{thm (lhs) Pos.simps(3)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(3)}\\
-  @{thm (lhs) Pos.simps(4)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(4)}\\
-  @{thm (lhs) Pos.simps(5)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
-  & \<open>\<equiv>\<close> 
-  & @{thm (rhs) Pos.simps(5)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
-  @{thm (lhs) Pos_stars} & \<open>\<equiv>\<close> & @{thm (rhs) Pos_stars}\\
-  \end{tabular}
-  \end{center}
-
-  \noindent 
-  whereby \<open>len\<close> in the last clause stands for the length of a list. Clearly
-  for every position inside a value there exists a subvalue at that position.
- 
-
-  To help understanding the ordering of Okui and Suzuki, consider again 
-  the earlier value
-  \<open>v\<close> and compare it with the following \<open>w\<close>:
-
-  \begin{center}
-  \begin{tabular}{l}
-  @{term "v == Stars [Seq (Char x) (Char y), Char z]"}\\
-  @{term "w == Stars [Char x, Char y, Char z]"}  
-  \end{tabular}
-  \end{center}
-
-  \noindent Both values match the string \<open>xyz\<close>, that means if
-  we flatten these values at their respective root position, we obtain
-  \<open>xyz\<close>. However, at position \<open>[0]\<close>, \<open>v\<close> matches
-  \<open>xy\<close> whereas \<open>w\<close> matches only the shorter \<open>x\<close>. So
-  according to the Longest Match Rule, we should prefer \<open>v\<close>,
-  rather than \<open>w\<close> as POSIX value for string \<open>xyz\<close> (and
-  corresponding regular expression). In order to
-  formalise this idea, Okui and Suzuki introduce a measure for
-  subvalues at position \<open>p\<close>, called the \emph{norm} of \<open>v\<close>
-  at position \<open>p\<close>. We can define this measure in Isabelle as an
-  integer as follows
-  
-  \begin{center}
-  @{thm pflat_len_def}
-  \end{center}
-
-  \noindent where we take the length of the flattened value at
-  position \<open>p\<close>, provided the position is inside \<open>v\<close>; if
-  not, then the norm is \<open>-1\<close>. The default for outside
-  positions is crucial for the POSIX requirement of preferring a
-  \<open>Left\<close>-value over a \<open>Right\<close>-value (if they can match the
-  same string---see the Priority Rule from the Introduction). For this
-  consider
-
-  \begin{center}
-  @{term "v == Left (Char x)"} \qquad and \qquad @{term "w == Right (Char x)"}
-  \end{center}
-
-  \noindent Both values match \<open>x\<close>. At position \<open>[0]\<close>
-  the norm of @{term v} is \<open>1\<close> (the subvalue matches \<open>x\<close>),
-  but the norm of \<open>w\<close> is \<open>-1\<close> (the position is outside
-  \<open>w\<close> according to how we defined the `inside' positions of
-  \<open>Left\<close>- and \<open>Right\<close>-values).  Of course at position
-  \<open>[1]\<close>, the norms @{term "pflat_len v [1]"} and @{term
-  "pflat_len w [1]"} are reversed, but the point is that subvalues
-  will be analysed according to lexicographically ordered
-  positions. According to this ordering, the position \<open>[0]\<close>
-  takes precedence over \<open>[1]\<close> and thus also \<open>v\<close> will be 
-  preferred over \<open>w\<close>.  The lexicographic ordering of positions, written
-  @{term "DUMMY \<sqsubset>lex DUMMY"}, can be conveniently formalised
-  by three inference rules
-
-  \begin{center}
-  \begin{tabular}{ccc}
-  @{thm [mode=Axiom] lex_list.intros(1)}\hspace{1cm} &
-  @{thm [mode=Rule] lex_list.intros(3)[where ?p1.0="p\<^sub>1" and ?p2.0="p\<^sub>2" and
-                                            ?ps1.0="ps\<^sub>1" and ?ps2.0="ps\<^sub>2"]}\hspace{1cm} &
-  @{thm [mode=Rule] lex_list.intros(2)[where ?ps1.0="ps\<^sub>1" and ?ps2.0="ps\<^sub>2"]}
-  \end{tabular}
-  \end{center}
-
-  With the norm and lexicographic order in place,
-  we can state the key definition of Okui and Suzuki
-  \cite{OkuiSuzuki2010}: a value @{term "v\<^sub>1"} is \emph{smaller at position \<open>p\<close>} than
-  @{term "v\<^sub>2"}, written @{term "v\<^sub>1 \<sqsubset>val p v\<^sub>2"}, 
-  if and only if  $(i)$ the norm at position \<open>p\<close> is
-  greater in @{term "v\<^sub>1"} (that is the string @{term "flat (at v\<^sub>1 p)"} is longer 
-  than @{term "flat (at v\<^sub>2 p)"}) and $(ii)$ all subvalues at 
-  positions that are inside @{term "v\<^sub>1"} or @{term "v\<^sub>2"} and that are
-  lexicographically smaller than \<open>p\<close>, we have the same norm, namely
-
- \begin{center}
- \begin{tabular}{c}
- @{thm (lhs) PosOrd_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} 
- \<open>\<equiv>\<close> 
- $\begin{cases}
- (i) & @{term "pflat_len v\<^sub>1 p > pflat_len v\<^sub>2 p"}   \quad\text{and}\smallskip \\
- (ii) & @{term "(\<forall>q \<in> Pos v\<^sub>1 \<union> Pos v\<^sub>2. q \<sqsubset>lex p --> pflat_len v\<^sub>1 q = pflat_len v\<^sub>2 q)"}
- \end{cases}$
- \end{tabular}
- \end{center}
-
- \noindent The position \<open>p\<close> in this definition acts as the
-  \emph{first distinct position} of \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close>, where both values match strings of different length
-  \cite{OkuiSuzuki2010}.  Since at \<open>p\<close> the values \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close> match different strings, the
-  ordering is irreflexive. Derived from the definition above
-  are the following two orderings:
-  
-  \begin{center}
-  \begin{tabular}{l}
-  @{thm PosOrd_ex_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
-  @{thm PosOrd_ex_eq_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
-  \end{tabular}
-  \end{center}
-
- While we encountered a number of obstacles for establishing properties like
- transitivity for the ordering of Sulzmann and Lu (and which we failed
- to overcome), it is relatively straightforward to establish this
- property for the orderings
- @{term "DUMMY :\<sqsubset>val DUMMY"} and @{term "DUMMY :\<sqsubseteq>val DUMMY"}  
- by Okui and Suzuki.
-
- \begin{lemma}[Transitivity]\label{transitivity}
- @{thm [mode=IfThen] PosOrd_trans[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and ?v3.0="v\<^sub>3"]} 
- \end{lemma}
-
- \begin{proof} From the assumption we obtain two positions \<open>p\<close>
- and \<open>q\<close>, where the values \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close> (respectively \<open>v\<^sub>2\<close> and \<open>v\<^sub>3\<close>) are `distinct'.  Since \<open>\<prec>\<^bsub>lex\<^esub>\<close> is trichotomous, we need to consider
- three cases, namely @{term "p = q"}, @{term "p \<sqsubset>lex q"} and
- @{term "q \<sqsubset>lex p"}. Let us look at the first case.  Clearly
- @{term "pflat_len v\<^sub>2 p < pflat_len v\<^sub>1 p"} and @{term
- "pflat_len v\<^sub>3 p < pflat_len v\<^sub>2 p"} imply @{term
- "pflat_len v\<^sub>3 p < pflat_len v\<^sub>1 p"}.  It remains to show
- that for a @{term "p' \<in> Pos v\<^sub>1 \<union> Pos v\<^sub>3"}
- with @{term "p' \<sqsubset>lex p"} that @{term "pflat_len v\<^sub>1
- p' = pflat_len v\<^sub>3 p'"} holds.  Suppose @{term "p' \<in> Pos
- v\<^sub>1"}, then we can infer from the first assumption that @{term
- "pflat_len v\<^sub>1 p' = pflat_len v\<^sub>2 p'"}.  But this means
- that @{term "p'"} must be in @{term "Pos v\<^sub>2"} too (the norm
- cannot be \<open>-1\<close> given @{term "p' \<in> Pos v\<^sub>1"}).  
- Hence we can use the second assumption and
- infer @{term "pflat_len v\<^sub>2 p' = pflat_len v\<^sub>3 p'"},
- which concludes this case with @{term "v\<^sub>1 :\<sqsubset>val
- v\<^sub>3"}.  The reasoning in the other cases is similar.\qed
- \end{proof}
-
- \noindent 
- The proof for $\preccurlyeq$ is similar and omitted.
- It is also straightforward to show that \<open>\<prec>\<close> and
- $\preccurlyeq$ are partial orders.  Okui and Suzuki furthermore show that they
- are linear orderings for lexical values \cite{OkuiSuzuki2010} of a given
- regular expression and given string, but we have not formalised this in Isabelle. It is
- not essential for our results. What we are going to show below is
- that for a given \<open>r\<close> and \<open>s\<close>, the orderings have a unique
- minimal element on the set @{term "LV r s"}, which is the POSIX value
- we defined in the previous section. We start with two properties that
- show how the length of a flattened value relates to the \<open>\<prec>\<close>-ordering.
-
- \begin{proposition}\mbox{}\smallskip\\\label{ordlen}
- \begin{tabular}{@ {}ll}
- (1) &
- @{thm [mode=IfThen] PosOrd_shorterE[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
- (2) &
- @{thm [mode=IfThen] PosOrd_shorterI[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} 
- \end{tabular} 
- \end{proposition}
- 
- \noindent Both properties follow from the definition of the ordering. Note that
- \textit{(2)} entails that a value, say @{term "v\<^sub>2"}, whose underlying 
- string is a strict prefix of another flattened value, say @{term "v\<^sub>1"}, then
- @{term "v\<^sub>1"} must be smaller than @{term "v\<^sub>2"}. For our proofs it
- will be useful to have the following properties---in each case the underlying strings 
- of the compared values are the same: 
-
-  \begin{proposition}\mbox{}\smallskip\\\label{ordintros}
-  \begin{tabular}{ll}
-  \textit{(1)} & 
-  @{thm [mode=IfThen] PosOrd_Left_Right[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
-  \textit{(2)} & If
-  @{thm (prem 1) PosOrd_Left_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \;then\;
-  @{thm (lhs) PosOrd_Left_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \;iff\;
-  @{thm (rhs) PosOrd_Left_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
-  \textit{(3)} & If
-  @{thm (prem 1) PosOrd_Right_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \;then\;
-  @{thm (lhs) PosOrd_Right_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \;iff\;
-  @{thm (rhs) PosOrd_Right_eq[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
-  \textit{(4)} & If
-  @{thm (prem 1) PosOrd_Seq_eq[where ?v2.0="v\<^sub>2" and ?w2.0="w\<^sub>2"]} \;then\;
-  @{thm (lhs) PosOrd_Seq_eq[where ?v2.0="v\<^sub>2" and ?w2.0="w\<^sub>2"]} \;iff\;
-  @{thm (rhs) PosOrd_Seq_eq[where ?v2.0="v\<^sub>2" and ?w2.0="w\<^sub>2"]}\\
-  \textit{(5)} & If
-  @{thm (prem 2) PosOrd_SeqI1[simplified, where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
-                                    ?w1.0="w\<^sub>1" and ?w2.0="w\<^sub>2"]} \;and\;
-  @{thm (prem 1) PosOrd_SeqI1[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
-                                    ?w1.0="w\<^sub>1" and ?w2.0="w\<^sub>2"]} \;then\;
-  @{thm (concl) PosOrd_SeqI1[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
-                                   ?w1.0="w\<^sub>1" and ?w2.0="w\<^sub>2"]}\\
-  \textit{(6)} & If
-  @{thm (prem 1) PosOrd_Stars_append_eq[where ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]} \;then\;
-  @{thm (lhs) PosOrd_Stars_append_eq[where ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]} \;iff\;
-  @{thm (rhs) PosOrd_Stars_append_eq[where ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]}\\  
-  
-  \textit{(7)} & If
-  @{thm (prem 2) PosOrd_StarsI[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
-                            ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]} \;and\;
-  @{thm (prem 1) PosOrd_StarsI[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
-                            ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]} \;then\;
-   @{thm (concl) PosOrd_StarsI[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and
-                            ?vs1.0="vs\<^sub>1" and ?vs2.0="vs\<^sub>2"]}\\
-  \end{tabular} 
-  \end{proposition}
-
-  \noindent One might prefer that statements \textit{(4)} and \textit{(5)} 
-  (respectively \textit{(6)} and \textit{(7)})
-  are combined into a single \textit{iff}-statement (like the ones for \<open>Left\<close> and \<open>Right\<close>). Unfortunately this cannot be done easily: such
-  a single statement would require an additional assumption about the
-  two values @{term "Seq v\<^sub>1 v\<^sub>2"} and @{term "Seq w\<^sub>1 w\<^sub>2"}
-  being inhabited by the same regular expression. The
-  complexity of the proofs involved seems to not justify such a
-  `cleaner' single statement. The statements given are just the properties that
-  allow us to establish our theorems without any difficulty. The proofs 
-  for Proposition~\ref{ordintros} are routine.
- 
-
-  Next we establish how Okui and Suzuki's orderings relate to our
-  definition of POSIX values.  Given a \<open>POSIX\<close> value \<open>v\<^sub>1\<close>
-  for \<open>r\<close> and \<open>s\<close>, then any other lexical value \<open>v\<^sub>2\<close> in @{term "LV r s"} is greater or equal than \<open>v\<^sub>1\<close>, namely:
-
-
-  \begin{theorem}\label{orderone}
-  @{thm [mode=IfThen] Posix_PosOrd[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
-  \end{theorem}
-
-  \begin{proof} By induction on our POSIX rules. By
-  Theorem~\ref{posixdeterm} and the definition of @{const LV}, it is clear
-  that \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close> have the same
-  underlying string @{term s}.  The three base cases are
-  straightforward: for example for @{term "v\<^sub>1 = Void"}, we have
-  that @{term "v\<^sub>2 \<in> LV ONE []"} must also be of the form
-  \mbox{@{term "v\<^sub>2 = Void"}}. Therefore we have @{term
-  "v\<^sub>1 :\<sqsubseteq>val v\<^sub>2"}.  The inductive cases for
-  \<open>r\<close> being of the form @{term "ALT r\<^sub>1 r\<^sub>2"} and
-  @{term "SEQ r\<^sub>1 r\<^sub>2"} are as follows:
-
-
-  \begin{itemize} 
-
-  \item[$\bullet$] Case \<open>P+L\<close> with @{term "s \<in> (ALT r\<^sub>1 r\<^sub>2)
-  \<rightarrow> (Left w\<^sub>1)"}: In this case the value 
-  @{term "v\<^sub>2"} is either of the
-  form @{term "Left w\<^sub>2"} or @{term "Right w\<^sub>2"}. In the
-  latter case we can immediately conclude with \mbox{@{term "v\<^sub>1
-  :\<sqsubseteq>val v\<^sub>2"}} since a \<open>Left\<close>-value with the
-  same underlying string \<open>s\<close> is always smaller than a
-  \<open>Right\<close>-value by Proposition~\ref{ordintros}\textit{(1)}.  
-  In the former case we have @{term "w\<^sub>2
-  \<in> LV r\<^sub>1 s"} and can use the induction hypothesis to infer
-  @{term "w\<^sub>1 :\<sqsubseteq>val w\<^sub>2"}. Because @{term
-  "w\<^sub>1"} and @{term "w\<^sub>2"} have the same underlying string
-  \<open>s\<close>, we can conclude with @{term "Left w\<^sub>1
-  :\<sqsubseteq>val Left w\<^sub>2"} using
-  Proposition~\ref{ordintros}\textit{(2)}.\smallskip
-
-  \item[$\bullet$] Case \<open>P+R\<close> with @{term "s \<in> (ALT r\<^sub>1 r\<^sub>2)
-  \<rightarrow> (Right w\<^sub>1)"}: This case similar to the previous
-  case, except that we additionally know @{term "s \<notin> L
-  r\<^sub>1"}. This is needed when @{term "v\<^sub>2"} is of the form
-  \mbox{@{term "Left w\<^sub>2"}}. Since \mbox{@{term "flat v\<^sub>2 = flat
-  w\<^sub>2"} \<open>= s\<close>} and @{term "\<Turnstile> w\<^sub>2 :
-  r\<^sub>1"}, we can derive a contradiction for \mbox{@{term "s \<notin> L
-  r\<^sub>1"}} using
-  Proposition~\ref{inhabs}. So also in this case \mbox{@{term "v\<^sub>1
-  :\<sqsubseteq>val v\<^sub>2"}}.\smallskip
-
-  \item[$\bullet$] Case \<open>PS\<close> with @{term "(s\<^sub>1 @
-  s\<^sub>2) \<in> (SEQ r\<^sub>1 r\<^sub>2) \<rightarrow> (Seq
-  w\<^sub>1 w\<^sub>2)"}: We can assume @{term "v\<^sub>2 = Seq
-  (u\<^sub>1) (u\<^sub>2)"} with @{term "\<Turnstile> u\<^sub>1 :
-  r\<^sub>1"} and \mbox{@{term "\<Turnstile> u\<^sub>2 :
-  r\<^sub>2"}}. We have @{term "s\<^sub>1 @ s\<^sub>2 = (flat
-  u\<^sub>1) @ (flat u\<^sub>2)"}.  By the side-condition of the
-  \<open>PS\<close>-rule we know that either @{term "s\<^sub>1 = flat
-  u\<^sub>1"} or that @{term "flat u\<^sub>1"} is a strict prefix of
-  @{term "s\<^sub>1"}. In the latter case we can infer @{term
-  "w\<^sub>1 :\<sqsubset>val u\<^sub>1"} by
-  Proposition~\ref{ordlen}\textit{(2)} and from this @{term "v\<^sub>1
-  :\<sqsubseteq>val v\<^sub>2"} by Proposition~\ref{ordintros}\textit{(5)}
-  (as noted above @{term "v\<^sub>1"} and @{term "v\<^sub>2"} must have the
-  same underlying string).
-  In the former case we know
-  @{term "u\<^sub>1 \<in> LV r\<^sub>1 s\<^sub>1"} and @{term
-  "u\<^sub>2 \<in> LV r\<^sub>2 s\<^sub>2"}. With this we can use the
-  induction hypotheses to infer @{term "w\<^sub>1 :\<sqsubseteq>val
-  u\<^sub>1"} and @{term "w\<^sub>2 :\<sqsubseteq>val u\<^sub>2"}. By
-  Proposition~\ref{ordintros}\textit{(4,5)} we can again infer 
-  @{term "v\<^sub>1 :\<sqsubseteq>val
-  v\<^sub>2"}.
-
-  \end{itemize}
-
-  \noindent The case for \<open>P\<star>\<close> is similar to the \<open>PS\<close>-case and omitted.\qed
-  \end{proof}
-
-  \noindent This theorem shows that our \<open>POSIX\<close> value for a
-  regular expression \<open>r\<close> and string @{term s} is in fact a
-  minimal element of the values in \<open>LV r s\<close>. By
-  Proposition~\ref{ordlen}\textit{(2)} we also know that any value in
-  \<open>LV r s'\<close>, with @{term "s'"} being a strict prefix, cannot be
-  smaller than \<open>v\<^sub>1\<close>. The next theorem shows the
-  opposite---namely any minimal element in @{term "LV r s"} must be a
-  \<open>POSIX\<close> value. This can be established by induction on \<open>r\<close>, but the proof can be drastically simplified by using the fact
-  from the previous section about the existence of a \<open>POSIX\<close> value
-  whenever a string @{term "s \<in> L r"}.
-
-
-  \begin{theorem}
-  @{thm [mode=IfThen] PosOrd_Posix[where ?v1.0="v\<^sub>1"]} 
-  \end{theorem}
-
-  \begin{proof} 
-  If @{thm (prem 1) PosOrd_Posix[where ?v1.0="v\<^sub>1"]} then 
-  @{term "s \<in> L r"} by Proposition~\ref{inhabs}. Hence by Theorem~\ref{lexercorrect}(2) 
-  there exists a
-  \<open>POSIX\<close> value @{term "v\<^sub>P"} with @{term "s \<in> r \<rightarrow> v\<^sub>P"}
-  and by Lemma~\ref{LVposix} we also have \mbox{@{term "v\<^sub>P \<in> LV r s"}}.
-  By Theorem~\ref{orderone} we therefore have 
-  @{term "v\<^sub>P :\<sqsubseteq>val v\<^sub>1"}. If @{term "v\<^sub>P = v\<^sub>1"} then
-  we are done. Otherwise we have @{term "v\<^sub>P :\<sqsubset>val v\<^sub>1"}, which 
-  however contradicts the second assumption about @{term "v\<^sub>1"} being the smallest
-  element in @{term "LV r s"}. So we are done in this case too.\qed
-  \end{proof}
-
-  \noindent
-  From this we can also show 
-  that if @{term "LV r s"} is non-empty (or equivalently @{term "s \<in> L r"}) then 
-  it has a unique minimal element:
-
-  \begin{corollary}
-  @{thm [mode=IfThen] Least_existence1}
-  \end{corollary}
-
-
-
-  \noindent To sum up, we have shown that the (unique) minimal elements 
-  of the ordering by Okui and Suzuki are exactly the \<open>POSIX\<close>
-  values we defined inductively in Section~\ref{posixsec}. This provides
-  an independent confirmation that our ternary relation formalises the
-  informal POSIX rules. 
-
-\<close>
-
-section \<open>Bitcoded Lexing\<close>
-
-
-
-
-text \<open>
-
-Incremental calculation of the value. To simplify the proof we first define the function
-@{const flex} which calculates the ``iterated'' injection function. With this we can 
-rewrite the lexer as
-
-\begin{center}
-@{thm lexer_flex}
-\end{center}
-
-
-\<close>
-
-section \<open>Optimisations\<close>
-
-text \<open>
-
-  Derivatives as calculated by \Brz's method are usually more complex
-  regular expressions than the initial one; the result is that the
-  derivative-based matching and lexing algorithms are often abysmally slow.
-  However, various optimisations are possible, such as the simplifications
-  of @{term "ALT ZERO r"}, @{term "ALT r ZERO"}, @{term "SEQ ONE r"} and
-  @{term "SEQ r ONE"} to @{term r}. These simplifications can speed up the
-  algorithms considerably, as noted in \cite{Sulzmann2014}. One of the
-  advantages of having a simple specification and correctness proof is that
-  the latter can be refined to prove the correctness of such simplification
-  steps. While the simplification of regular expressions according to 
-  rules like
-
-  \begin{equation}\label{Simpl}
-  \begin{array}{lcllcllcllcl}
-  @{term "ALT ZERO r"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
-  @{term "ALT r ZERO"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
-  @{term "SEQ ONE r"}  & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
-  @{term "SEQ r ONE"}  & \<open>\<Rightarrow>\<close> & @{term r}
-  \end{array}
-  \end{equation}
-
-  \noindent is well understood, there is an obstacle with the POSIX value
-  calculation algorithm by Sulzmann and Lu: if we build a derivative regular
-  expression and then simplify it, we will calculate a POSIX value for this
-  simplified derivative regular expression, \emph{not} for the original (unsimplified)
-  derivative regular expression. Sulzmann and Lu \cite{Sulzmann2014} overcome this obstacle by
-  not just calculating a simplified regular expression, but also calculating
-  a \emph{rectification function} that ``repairs'' the incorrect value.
-  
-  The rectification functions can be (slightly clumsily) implemented  in
-  Isabelle/HOL as follows using some auxiliary functions:
-
-  \begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) F_RIGHT.simps(1)} & $\dn$ & \<open>Right (f v)\<close>\\
-  @{thm (lhs) F_LEFT.simps(1)} & $\dn$ & \<open>Left (f v)\<close>\\
-  
-  @{thm (lhs) F_ALT.simps(1)} & $\dn$ & \<open>Right (f\<^sub>2 v)\<close>\\
-  @{thm (lhs) F_ALT.simps(2)} & $\dn$ & \<open>Left (f\<^sub>1 v)\<close>\\
-  
-  @{thm (lhs) F_SEQ1.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 ()) (f\<^sub>2 v)\<close>\\
-  @{thm (lhs) F_SEQ2.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 v) (f\<^sub>2 ())\<close>\\
-  @{thm (lhs) F_SEQ.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 v\<^sub>1) (f\<^sub>2 v\<^sub>2)\<close>\medskip\\
-  %\end{tabular}
-  %
-  %\begin{tabular}{lcl}
-  @{term "simp_ALT (ZERO, DUMMY) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>2, F_RIGHT f\<^sub>2)"}\\
-  @{term "simp_ALT (r\<^sub>1, f\<^sub>1) (ZERO, DUMMY)"} & $\dn$ & @{term "(r\<^sub>1, F_LEFT f\<^sub>1)"}\\
-  @{term "simp_ALT (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(ALT r\<^sub>1 r\<^sub>2, F_ALT f\<^sub>1 f\<^sub>2)"}\\
-  @{term "simp_SEQ (ONE, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>2, F_SEQ1 f\<^sub>1 f\<^sub>2)"}\\
-  @{term "simp_SEQ (r\<^sub>1, f\<^sub>1) (ONE, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>1, F_SEQ2 f\<^sub>1 f\<^sub>2)"}\\
-  @{term "simp_SEQ (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(SEQ r\<^sub>1 r\<^sub>2, F_SEQ f\<^sub>1 f\<^sub>2)"}\\
-  \end{tabular}
-  \end{center}
-
-  \noindent
-  The functions \<open>simp\<^bsub>Alt\<^esub>\<close> and \<open>simp\<^bsub>Seq\<^esub>\<close> encode the simplification rules
-  in \eqref{Simpl} and compose the rectification functions (simplifications can occur
-  deep inside the regular expression). The main simplification function is then 
-
-  \begin{center}
-  \begin{tabular}{lcl}
-  @{term "simp (ALT r\<^sub>1 r\<^sub>2)"} & $\dn$ & @{term "simp_ALT (simp r\<^sub>1) (simp r\<^sub>2)"}\\
-  @{term "simp (SEQ r\<^sub>1 r\<^sub>2)"} & $\dn$ & @{term "simp_SEQ (simp r\<^sub>1) (simp r\<^sub>2)"}\\
-  @{term "simp r"} & $\dn$ & @{term "(r, id)"}\\
-  \end{tabular}
-  \end{center} 
-
-  \noindent where @{term "id"} stands for the identity function. The
-  function @{const simp} returns a simplified regular expression and a corresponding
-  rectification function. Note that we do not simplify under stars: this
-  seems to slow down the algorithm, rather than speed it up. The optimised
-  lexer is then given by the clauses:
-  
-  \begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) slexer.simps(1)} & $\dn$ & @{thm (rhs) slexer.simps(1)}\\
-  @{thm (lhs) slexer.simps(2)} & $\dn$ & 
-                         \<open>let (r\<^sub>s, f\<^sub>r) = simp (r \<close>$\backslash$\<open> c) in\<close>\\
-                     & & \<open>case\<close> @{term "slexer r\<^sub>s s"} \<open>of\<close>\\
-                     & & \phantom{$|$} @{term "None"}  \<open>\<Rightarrow>\<close> @{term None}\\
-                     & & $|$ @{term "Some v"} \<open>\<Rightarrow>\<close> \<open>Some (inj r c (f\<^sub>r v))\<close>                          
-  \end{tabular}
-  \end{center}
-
-  \noindent
-  In the second clause we first calculate the derivative @{term "der c r"}
-  and then simpli
-
-text \<open>
-
-Incremental calculation of the value. To simplify the proof we first define the function
-@{const flex} which calculates the ``iterated'' injection function. With this we can 
-rewrite the lexer as
-
-\begin{center}
-@{thm lexer_flex}
-\end{center}
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) code.simps(1)} & $\dn$ & @{thm (rhs) code.simps(1)}\\
-  @{thm (lhs) code.simps(2)} & $\dn$ & @{thm (rhs) code.simps(2)}\\
-  @{thm (lhs) code.simps(3)} & $\dn$ & @{thm (rhs) code.simps(3)}\\
-  @{thm (lhs) code.simps(4)} & $\dn$ & @{thm (rhs) code.simps(4)}\\
-  @{thm (lhs) code.simps(5)[of "v\<^sub>1" "v\<^sub>2"]} & $\dn$ & @{thm (rhs) code.simps(5)[of "v\<^sub>1" "v\<^sub>2"]}\\
-  @{thm (lhs) code.simps(6)} & $\dn$ & @{thm (rhs) code.simps(6)}\\
-  @{thm (lhs) code.simps(7)} & $\dn$ & @{thm (rhs) code.simps(7)}
-\end{tabular}
-\end{center}
-
-\begin{center}
-\begin{tabular}{lcl}
-  @{term areg} & $::=$ & @{term "AZERO"}\\
-               & $\mid$ & @{term "AONE bs"}\\
-               & $\mid$ & @{term "ACHAR bs c"}\\
-               & $\mid$ & @{term "AALT bs r1 r2"}\\
-               & $\mid$ & @{term "ASEQ bs r\<^sub>1 r\<^sub>2"}\\
-               & $\mid$ & @{term "ASTAR bs r"}
-\end{tabular}
-\end{center}
-
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) intern.simps(1)} & $\dn$ & @{thm (rhs) intern.simps(1)}\\
-  @{thm (lhs) intern.simps(2)} & $\dn$ & @{thm (rhs) intern.simps(2)}\\
-  @{thm (lhs) intern.simps(3)} & $\dn$ & @{thm (rhs) intern.simps(3)}\\
-  @{thm (lhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) intern.simps(6)} & $\dn$ & @{thm (rhs) intern.simps(6)}\\
-\end{tabular}
-\end{center}
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) erase.simps(1)} & $\dn$ & @{thm (rhs) erase.simps(1)}\\
-  @{thm (lhs) erase.simps(2)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(2)[of bs]}\\
-  @{thm (lhs) erase.simps(3)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(3)[of bs]}\\
-  @{thm (lhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) erase.simps(6)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(6)[of bs]}\\
-\end{tabular}
-\end{center}
-
-Some simple facts about erase
-
-\begin{lemma}\mbox{}\\
-@{thm erase_bder}\\
-@{thm erase_intern}
-\end{lemma}
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) bnullable.simps(1)} & $\dn$ & @{thm (rhs) bnullable.simps(1)}\\
-  @{thm (lhs) bnullable.simps(2)} & $\dn$ & @{thm (rhs) bnullable.simps(2)}\\
-  @{thm (lhs) bnullable.simps(3)} & $\dn$ & @{thm (rhs) bnullable.simps(3)}\\
-  @{thm (lhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bnullable.simps(6)} & $\dn$ & @{thm (rhs) bnullable.simps(6)}\medskip\\
-
-%  \end{tabular}
-%  \end{center}
-
-%  \begin{center}
-%  \begin{tabular}{lcl}
-
-  @{thm (lhs) bder.simps(1)} & $\dn$ & @{thm (rhs) bder.simps(1)}\\
-  @{thm (lhs) bder.simps(2)} & $\dn$ & @{thm (rhs) bder.simps(2)}\\
-  @{thm (lhs) bder.simps(3)} & $\dn$ & @{thm (rhs) bder.simps(3)}\\
-  @{thm (lhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bder.simps(6)} & $\dn$ & @{thm (rhs) bder.simps(6)}
-  \end{tabular}
-  \end{center}
-
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) bmkeps.simps(1)} & $\dn$ & @{thm (rhs) bmkeps.simps(1)}\\
-  @{thm (lhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bmkeps.simps(4)} & $\dn$ & @{thm (rhs) bmkeps.simps(4)}\medskip\\
-\end{tabular}
-\end{center}
-
-
-@{thm [mode=IfThen] bder_retrieve}
-
-By induction on \<open>r\<close>
-
-\begin{theorem}[Main Lemma]\mbox{}\\
-@{thm [mode=IfThen] MAIN_decode}
-\end{theorem}
-
-\noindent
-Definition of the bitcoded lexer
-
-@{thm blexer_def}
-
-
-\begin{theorem}
-@{thm blexer_correctness}
-\end{theorem}
-
-\<close>
-
-section \<open>Optimisations\<close>
-
-text \<open>
-
-  Derivatives as calculated by \Brz's method are usually more complex
-  regular expressions than the initial one; the result is that the
-  derivative-based matching and lexing algorithms are often abysmally slow.
-  However, various optimisations are possible, such as the simplifications
-  of @{term "ALT ZERO r"}, @{term "ALT r ZERO"}, @{term "SEQ ONE r"} and
-  @{term "SEQ r ONE"} to @{term r}. These simplifications can speed up the
-  algorithms considerably, as noted in \cite{Sulzmann2014}. One of the
-  advantages of having a simple specification and correctness proof is that
-  the latter can be refined to prove the correctness of such simplification
-  steps. While the simplification of regular expressions according to 
-  rules like
-
-  \begin{equation}\label{Simpl}
-  \begin{array}{lcllcllcllcl}
-  @{term "ALT ZERO r"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
-  @{term "ALT r ZERO"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
-  @{term "SEQ ONE r"}  & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
-  @{term "SEQ r ONE"}  & \<open>\<Rightarrow>\<close> & @{term r}
-  \end{array}
-  \end{equation}
-
-  \noindent is well understood, there is an obstacle with the POSIX value
-  calculation algorithm by Sulzmann and Lu: if we build a derivative regular
-  expression and then simplify it, we will calculate a POSIX value for this
-  simplified derivative regular expression, \emph{not} for the original (unsimplified)
-  derivative regular expression. Sulzmann and Lu \cite{Sulzmann2014} overcome this obstacle by
-  not just calculating a simplified regular expression, but also calculating
-  a \emph{rectification function} that ``repairs'' the incorrect value.
-  
-  The rectification functions can be (slightly clumsily) implemented  in
-  Isabelle/HOL as follows using some auxiliary functions:
-
-  \begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) F_RIGHT.simps(1)} & $\dn$ & \<open>Right (f v)\<close>\\
-  @{thm (lhs) F_LEFT.simps(1)} & $\dn$ & \<open>Left (f v)\<close>\\
-  
-  @{thm (lhs) F_ALT.simps(1)} & $\dn$ & \<open>Right (f\<^sub>2 v)\<close>\\
-  @{thm (lhs) F_ALT.simps(2)} & $\dn$ & \<open>Left (f\<^sub>1 v)\<close>\\
-  
-  @{thm (lhs) F_SEQ1.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 ()) (f\<^sub>2 v)\<close>\\
-  @{thm (lhs) F_SEQ2.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 v) (f\<^sub>2 ())\<close>\\
-  @{thm (lhs) F_SEQ.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 v\<^sub>1) (f\<^sub>2 v\<^sub>2)\<close>\medskip\\
-  %\end{tabular}
-  %
-  %\begin{tabular}{lcl}
-  @{term "simp_ALT (ZERO, DUMMY) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>2, F_RIGHT f\<^sub>2)"}\\
-  @{term "simp_ALT (r\<^sub>1, f\<^sub>1) (ZERO, DUMMY)"} & $\dn$ & @{term "(r\<^sub>1, F_LEFT f\<^sub>1)"}\\
-  @{term "simp_ALT (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(ALT r\<^sub>1 r\<^sub>2, F_ALT f\<^sub>1 f\<^sub>2)"}\\
-  @{term "simp_SEQ (ONE, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>2, F_SEQ1 f\<^sub>1 f\<^sub>2)"}\\
-  @{term "simp_SEQ (r\<^sub>1, f\<^sub>1) (ONE, f\<^sub>2)"} & $\dn$ & @{term "(r\<^sub>1, F_SEQ2 f\<^sub>1 f\<^sub>2)"}\\
-  @{term "simp_SEQ (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"} & $\dn$ & @{term "(SEQ r\<^sub>1 r\<^sub>2, F_SEQ f\<^sub>1 f\<^sub>2)"}\\
-  \end{tabular}
-  \end{center}
-
-  \noindent
-  The functions \<open>simp\<^bsub>Alt\<^esub>\<close> and \<open>simp\<^bsub>Seq\<^esub>\<close> encode the simplification rules
-  in \eqref{Simpl} and compose the rectification functions (simplifications can occur
-  deep inside the regular expression). The main simplification function is then 
-
-  \begin{center}
-  \begin{tabular}{lcl}
-  @{term "simp (ALT r\<^sub>1 r\<^sub>2)"} & $\dn$ & @{term "simp_ALT (simp r\<^sub>1) (simp r\<^sub>2)"}\\
-  @{term "simp (SEQ r\<^sub>1 r\<^sub>2)"} & $\dn$ & @{term "simp_SEQ (simp r\<^sub>1) (simp r\<^sub>2)"}\\
-  @{term "simp r"} & $\dn$ & @{term "(r, id)"}\\
-  \end{tabular}
-  \end{center} 
-
-  \noindent where @{term "id"} stands for the identity function. The
-  function @{const simp} returns a simplified regular expression and a corresponding
-  rectification function. Note that we do not simplify under stars: this
-  seems to slow down the algorithm, rather than speed it up. The optimised
-  lexer is then given by the clauses:
-  
-  \begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) slexer.simps(1)} & $\dn$ & @{thm (rhs) slexer.simps(1)}\\
-  @{thm (lhs) slexer.simps(2)} & $\dn$ & 
-                         \<open>let (r\<^sub>s, f\<^sub>r) = simp (r \<close>$\backslash$\<open> c) in\<close>\\
-                     & & \<open>case\<close> @{term "slexer r\<^sub>s s"} \<open>of\<close>\\
-                     & & \phantom{$|$} @{term "None"}  \<open>\<Rightarrow>\<close> @{term None}\\
-                     & & $|$ @{term "Some v"} \<open>\<Rightarrow>\<close> \<open>Some (inj r c (f\<^sub>r v))\<close>                          
-  \end{tabular}
-  \end{center}
-
-  \noindent
-  In the second clause we first calculate the derivative @{term "der c r"}
-  and then simplify the result. This gives us a simplified derivative
-  \<open>r\<^sub>s\<close> and a rectification function \<open>f\<^sub>r\<close>. The lexer
-  is then recursively called with the simplified derivative, but before
-  we inject the character @{term c} into the value @{term v}, we need to rectify
-  @{term v} (that is construct @{term "f\<^sub>r v"}). Before we can establish the correctness
-  of @{term "slexer"}, we need to show that simplification preserves the language
-  and simplification preserves our POSIX relation once the value is rectified
-  (recall @{const "simp"} generates a (regular expression, rectification function) pair):
-
-  \begin{lemma}\mbox{}\smallskip\\\label{slexeraux}
-  \begin{tabular}{ll}
-  (1) & @{thm L_fst_simp[symmetric]}\\
-  (2) & @{thm[mode=IfThen] Posix_simp}
-  \end{tabular}
-  \end{lemma}
-
-  \begin{proof} Both are by induction on \<open>r\<close>. There is no
-  interesting case for the first statement. For the second statement,
-  of interest are the @{term "r = ALT r\<^sub>1 r\<^sub>2"} and @{term "r = SEQ r\<^sub>1
-  r\<^sub>2"} cases. In each case we have to analyse four subcases whether
-  @{term "fst (simp r\<^sub>1)"} and @{term "fst (simp r\<^sub>2)"} equals @{const
-  ZERO} (respectively @{const ONE}). For example for @{term "r = ALT
-  r\<^sub>1 r\<^sub>2"}, consider the subcase @{term "fst (simp r\<^sub>1) = ZERO"} and
-  @{term "fst (simp r\<^sub>2) \<noteq> ZERO"}. By assumption we know @{term "s \<in>
-  fst (simp (ALT r\<^sub>1 r\<^sub>2)) \<rightarrow> v"}. From this we can infer @{term "s \<in> fst (simp r\<^sub>2) \<rightarrow> v"}
-  and by IH also (*) @{term "s \<in> r\<^sub>2 \<rightarrow> (snd (simp r\<^sub>2) v)"}. Given @{term "fst (simp r\<^sub>1) = ZERO"}
-  we know @{term "L (fst (simp r\<^sub>1)) = {}"}. By the first statement
-  @{term "L r\<^sub>1"} is the empty set, meaning (**) @{term "s \<notin> L r\<^sub>1"}.
-  Taking (*) and (**) together gives by the \mbox{\<open>P+R\<close>}-rule 
-  @{term "s \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> Right (snd (simp r\<^sub>2) v)"}. In turn this
-  gives @{term "s \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> snd (simp (ALT r\<^sub>1 r\<^sub>2)) v"} as we need to show.
-  The other cases are similar.\qed
-  \end{proof}
-
-  \noindent We can now prove relatively straightforwardly that the
-  optimised lexer produces the expected result:
-
-  \begin{theorem}
-  @{thm slexer_correctness}
-  \end{theorem}
-
-  \begin{proof} By induction on @{term s} generalising over @{term
-  r}. The case @{term "[]"} is trivial. For the cons-case suppose the
-  string is of the form @{term "c # s"}. By induction hypothesis we
-  know @{term "slexer r s = lexer r s"} holds for all @{term r} (in
-  particular for @{term "r"} being the derivative @{term "der c
-  r"}). Let @{term "r\<^sub>s"} be the simplified derivative regular expression, that is @{term
-  "fst (simp (der c r))"}, and @{term "f\<^sub>r"} be the rectification
-  function, that is @{term "snd (simp (der c r))"}.  We distinguish the cases
-  whether (*) @{term "s \<in> L (der c r)"} or not. In the first case we
-  have by Theorem~\ref{lexercorrect}(2) a value @{term "v"} so that @{term
-  "lexer (der c r) s = Some v"} and @{term "s \<in> der c r \<rightarrow> v"} hold.
-  By Lemma~\ref{slexeraux}(1) we can also infer from~(*) that @{term "s
-  \<in> L r\<^sub>s"} holds.  Hence we know by Theorem~\ref{lexercorrect}(2) that
-  there exists a @{term "v'"} with @{term "lexer r\<^sub>s s = Some v'"} and
-  @{term "s \<in> r\<^sub>s \<rightarrow> v'"}. From the latter we know by
-  Lemma~\ref{slexeraux}(2) that @{term "s \<in> der c r \<rightarrow> (f\<^sub>r v')"} holds.
-  By the uniqueness of the POSIX relation (Theorem~\ref{posixdeterm}) we
-  can infer that @{term v} is equal to @{term "f\<^sub>r v'"}---that is the 
-  rectification function applied to @{term "v'"}
-  produces the original @{term "v"}.  Now the case follows by the
-  definitions of @{const lexer} and @{const slexer}.
-
-  In the second case where @{term "s \<notin> L (der c r)"} we have that
-  @{term "lexer (der c r) s = None"} by Theorem~\ref{lexercorrect}(1).  We
-  also know by Lemma~\ref{slexeraux}(1) that @{term "s \<notin> L r\<^sub>s"}. Hence
-  @{term "lexer r\<^sub>s s = None"} by Theorem~\ref{lexercorrect}(1) and
-  by IH then also @{term "slexer r\<^sub>s s = None"}. With this we can
-  conclude in this case too.\qed   
-
-  \end{proof} 
-
-\<close>
-fy the result. This gives us a simplified derivative
-  \<open>r\<^sub>s\<close> and a rectification function \<open>f\<^sub>r\<close>. The lexer
-  is then recursively called with the simplified derivative, but before
-  we inject the character @{term c} into the value @{term v}, we need to rectify
-  @{term v} (that is construct @{term "f\<^sub>r v"}). Before we can establish the correctness
-  of @{term "slexer"}, we need to show that simplification preserves the language
-  and simplification preserves our POSIX relation once the value is rectified
-  (recall @{const "simp"} generates a (regular expression, rectification function) pair):
-
-  \begin{lemma}\mbox{}\smallskip\\\label{slexeraux}
-  \begin{tabular}{ll}
-  (1) & @{thm L_fst_simp[symmetric]}\\
-  (2) & @{thm[mode=IfThen] Posix_simp}
-  \end{tabular}
-  \end{lemma}
-
-  \begin{proof} Both are by induction on \<open>r\<close>. There is no
-  interesting case for the first statement. For the second statement,
-  of interest are the @{term "r = ALT r\<^sub>1 r\<^sub>2"} and @{term "r = SEQ r\<^sub>1
-  r\<^sub>2"} cases. In each case we have to analyse four subcases whether
-  @{term "fst (simp r\<^sub>1)"} and @{term "fst (simp r\<^sub>2)"} equals @{const
-  ZERO} (respectively @{const ONE}). For example for @{term "r = ALT
-  r\<^sub>1 r\<^sub>2"}, consider the subcase @{term "fst (simp r\<^sub>1) = ZERO"} and
-  @{term "fst (simp r\<^sub>2) \<noteq> ZERO"}. By assumption we know @{term "s \<in>
-  fst (simp (ALT r\<^sub>1 r\<^sub>2)) \<rightarrow> v"}. From this we can infer @{term "s \<in> fst (simp r\<^sub>2) \<rightarrow> v"}
-  and by IH also (*) @{term "s \<in> r\<^sub>2 \<rightarrow> (snd (simp r\<^sub>2) v)"}. Given @{term "fst (simp r\<^sub>1) = ZERO"}
-  we know @{term "L (fst (simp r\<^sub>1)) = {}"}. By the first statement
-  @{term "L r\<^sub>1"} is the empty set, meaning (**) @{term "s \<notin> L r\<^sub>1"}.
-  Taking (*) and (**) together gives by the \mbox{\<open>P+R\<close>}-rule 
-  @{term "s \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> Right (snd (simp r\<^sub>2) v)"}. In turn this
-  gives @{term "s \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> snd (simp (ALT r\<^sub>1 r\<^sub>2)) v"} as we need to show.
-  The other cases are similar.\qed
-  \end{proof}
-
-  \noindent We can now prove relatively straightforwardly that the
-  optimised lexer produces the expected result:
-
-  \begin{theorem}
-  @{thm slexer_correctness}
-  \end{theorem}
-
-  \begin{proof} By induction on @{term s} generalising over @{term
-  r}. The case @{term "[]"} is trivial. For the cons-case suppose the
-  string is of the form @{term "c # s"}. By induction hypothesis we
-  know @{term "slexer r s = lexer r s"} holds for all @{term r} (in
-  particular for @{term "r"} being the derivative @{term "der c
-  r"}). Let @{term "r\<^sub>s"} be the simplified derivative regular expression, that is @{term
-  "fst (simp (der c r))"}, and @{term "f\<^sub>r"} be the rectification
-  function, that is @{term "snd (simp (der c r))"}.  We distinguish the cases
-  whether (*) @{term "s \<in> L (der c r)"} or not. In the first case we
-  have by Theorem~\ref{lexercorrect}(2) a value @{term "v"} so that @{term
-  "lexer (der c r) s = Some v"} and @{term "s \<in> der c r \<rightarrow> v"} hold.
-  By Lemma~\ref{slexeraux}(1) we can also infer from~(*) that @{term "s
-  \<in> L r\<^sub>s"} holds.  Hence we know by Theorem~\ref{lexercorrect}(2) that
-  there exists a @{term "v'"} with @{term "lexer r\<^sub>s s = Some v'"} and
-  @{term "s \<in> r\<^sub>s \<rightarrow> v'"}. From the latter we know by
-  Lemma~\ref{slexeraux}(2) that @{term "s \<in> der c r \<rightarrow> (f\<^sub>r v')"} holds.
-  By the uniqueness of the POSIX relation (Theorem~\ref{posixdeterm}) we
-  can infer that @{term v} is equal to @{term "f\<^sub>r v'"}---that is the 
-  rectification function applied to @{term "v'"}
-  produces the original @{term "v"}.  Now the case follows by the
-  definitions of @{const lexer} and @{const slexer}.
-
-  In the second case where @{term "s \<notin> L (der c r)"} we have that
-  @{term "lexer (der c r) s = None"} by Theorem~\ref{lexercorrect}(1).  We
-  also know by Lemma~\ref{slexeraux}(1) that @{term "s \<notin> L r\<^sub>s"}. Hence
-  @{term "lexer r\<^sub>s s = None"} by Theorem~\ref{lexercorrect}(1) and
-  by IH then also @{term "slexer r\<^sub>s s = None"}. With this we can
-  conclude in this case too.\qed   
-
-  \end{proof} 
-
-\<close>
-
-
-section \<open>HERE\<close>
-
-text \<open>
-
-  \begin{lemma}
-  @{thm [mode=IfThen] bder_retrieve}
-  \end{lemma}
-
-  \begin{proof}
-  By induction on the definition of @{term "erase r"}. The cases for rule 1) and 2) are
-  straightforward as @{term "der c ZERO"} and @{term "der c ONE"} are both equal to 
-  @{term ZERO}. This means @{term "\<Turnstile> v : ZERO"} cannot hold. Similarly in case of rule 3)
-  where @{term r} is of the form @{term "ACHAR d"} with @{term "c = d"}. Then by assumption
-  we know @{term "\<Turnstile> v : ONE"}, which implies @{term "v = Void"}. The equation follows by 
-  simplification of left- and right-hand side. In  case @{term "c \<noteq> d"} we have again
-  @{term "\<Turnstile> v : ZERO"}, which cannot  hold. 
-
-  For rule 4a) we have again @{term "\<Turnstile> v : ZERO"}. The property holds by IH for rule 4b).
-  The  induction hypothesis is 
-  \[
-  @{term "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"}
-  \]
-  which is what left- and right-hand side simplify to.  The slightly more interesting case
-  is for 4c). By assumption  we have 
-  @{term "\<Turnstile> v : ALT (der c (erase r\<^sub>1)) (der c (erase (AALTs bs (r\<^sub>2 # rs))))"}. This means we 
-  have either (*) @{term "\<Turnstile> v1 : der c (erase r\<^sub>1)"} with @{term "v = Left v1"} or
-  (**) @{term "\<Turnstile> v2 : der c (erase (AALTs bs (r\<^sub>2 # rs)))"} with @{term "v = Right v2"}.
-  The former  case is straightforward by simplification. The second case is \ldots TBD.
-
-  Rule 5) TBD.
-
-  Finally for rule 6) the reasoning is as follows:   By assumption we  have
-  @{term "\<Turnstile> v : SEQ (der c (erase r)) (STAR (erase r))"}. This means we also have
-  @{term "v = Seq v1 v2"}, @{term "\<Turnstile> v1 : der c (erase r)"}  and @{term "v2 = Stars vs"}.
-  We want to prove
-  \begin{align}
-  & @{term "retrieve (ASEQ bs (fuse [Z] (bder c r)) (ASTAR [] r)) v"}\\
-  &= @{term "retrieve (ASTAR bs r) (injval (STAR (erase r)) c v)"}
-  \end{align}
-  The right-hand side @{term inj}-expression is equal to 
-  @{term "Stars (injval (erase r) c v1 # vs)"}, which means the @{term  retrieve}-expression
-  simplifies to 
-  \[
-  @{term "bs @ [Z] @ retrieve r (injval (erase r) c v1) @ retrieve (ASTAR [] r) (Stars vs)"}
-  \]
-  The left-hand side (3) above simplifies to 
-  \[
-  @{term "bs @ retrieve (fuse [Z] (bder c r)) v1 @ retrieve (ASTAR [] r) (Stars vs)"} 
-  \]
-  We can move out the @{term "fuse  [Z]"} and then use the IH to show that left-hand side
-  and right-hand side are equal. This completes the proof. 
-  \end{proof}   
-
-   
-
-  \bibliographystyle{plain}
-  \bibliography{root}
-
-\<close>
-(*<*)
-end
-(*>*)
-
-(*
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) code.simps(1)} & $\dn$ & @{thm (rhs) code.simps(1)}\\
-  @{thm (lhs) code.simps(2)} & $\dn$ & @{thm (rhs) code.simps(2)}\\
-  @{thm (lhs) code.simps(3)} & $\dn$ & @{thm (rhs) code.simps(3)}\\
-  @{thm (lhs) code.simps(4)} & $\dn$ & @{thm (rhs) code.simps(4)}\\
-  @{thm (lhs) code.simps(5)[of "v\<^sub>1" "v\<^sub>2"]} & $\dn$ & @{thm (rhs) code.simps(5)[of "v\<^sub>1" "v\<^sub>2"]}\\
-  @{thm (lhs) code.simps(6)} & $\dn$ & @{thm (rhs) code.simps(6)}\\
-  @{thm (lhs) code.simps(7)} & $\dn$ & @{thm (rhs) code.simps(7)}
-\end{tabular}
-\end{center}
-
-\begin{center}
-\begin{tabular}{lcl}
-  @{term areg} & $::=$ & @{term "AZERO"}\\
-               & $\mid$ & @{term "AONE bs"}\\
-               & $\mid$ & @{term "ACHAR bs c"}\\
-               & $\mid$ & @{term "AALT bs r1 r2"}\\
-               & $\mid$ & @{term "ASEQ bs r\<^sub>1 r\<^sub>2"}\\
-               & $\mid$ & @{term "ASTAR bs r"}
-\end{tabular}
-\end{center}
-
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) intern.simps(1)} & $\dn$ & @{thm (rhs) intern.simps(1)}\\
-  @{thm (lhs) intern.simps(2)} & $\dn$ & @{thm (rhs) intern.simps(2)}\\
-  @{thm (lhs) intern.simps(3)} & $\dn$ & @{thm (rhs) intern.simps(3)}\\
-  @{thm (lhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) intern.simps(6)} & $\dn$ & @{thm (rhs) intern.simps(6)}\\
-\end{tabular}
-\end{center}
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) erase.simps(1)} & $\dn$ & @{thm (rhs) erase.simps(1)}\\
-  @{thm (lhs) erase.simps(2)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(2)[of bs]}\\
-  @{thm (lhs) erase.simps(3)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(3)[of bs]}\\
-  @{thm (lhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) erase.simps(6)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(6)[of bs]}\\
-\end{tabular}
-\end{center}
-
-Some simple facts about erase
-
-\begin{lemma}\mbox{}\\
-@{thm erase_bder}\\
-@{thm erase_intern}
-\end{lemma}
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) bnullable.simps(1)} & $\dn$ & @{thm (rhs) bnullable.simps(1)}\\
-  @{thm (lhs) bnullable.simps(2)} & $\dn$ & @{thm (rhs) bnullable.simps(2)}\\
-  @{thm (lhs) bnullable.simps(3)} & $\dn$ & @{thm (rhs) bnullable.simps(3)}\\
-  @{thm (lhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bnullable.simps(6)} & $\dn$ & @{thm (rhs) bnullable.simps(6)}\medskip\\
-
-%  \end{tabular}
-%  \end{center}
-
-%  \begin{center}
-%  \begin{tabular}{lcl}
-
-  @{thm (lhs) bder.simps(1)} & $\dn$ & @{thm (rhs) bder.simps(1)}\\
-  @{thm (lhs) bder.simps(2)} & $\dn$ & @{thm (rhs) bder.simps(2)}\\
-  @{thm (lhs) bder.simps(3)} & $\dn$ & @{thm (rhs) bder.simps(3)}\\
-  @{thm (lhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bder.simps(6)} & $\dn$ & @{thm (rhs) bder.simps(6)}
-  \end{tabular}
-  \end{center}
-
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) bmkeps.simps(1)} & $\dn$ & @{thm (rhs) bmkeps.simps(1)}\\
-  @{thm (lhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bmkeps.simps(4)} & $\dn$ & @{thm (rhs) bmkeps.simps(4)}\medskip\\
-\end{tabular}
-\end{center}
-
-
-@{thm [mode=IfThen] bder_retrieve}
-
-By induction on \<open>r\<close>
-
-\begin{theorem}[Main Lemma]\mbox{}\\
-@{thm [mode=IfThen] MAIN_decode}
-\end{theorem}
-
-\noindent
-Definition of the bitcoded lexer
-
-@{thm blexer_def}
-
-
-\begin{theorem}
-@{thm blexer_correctness}
-\end{theorem}
-
-
-
-
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) code.simps(1)} & $\dn$ & @{thm (rhs) code.simps(1)}\\
-  @{thm (lhs) code.simps(2)} & $\dn$ & @{thm (rhs) code.simps(2)}\\
-  @{thm (lhs) code.simps(3)} & $\dn$ & @{thm (rhs) code.simps(3)}\\
-  @{thm (lhs) code.simps(4)} & $\dn$ & @{thm (rhs) code.simps(4)}\\
-  @{thm (lhs) code.simps(5)[of "v\<^sub>1" "v\<^sub>2"]} & $\dn$ & @{thm (rhs) code.simps(5)[of "v\<^sub>1" "v\<^sub>2"]}\\
-  @{thm (lhs) code.simps(6)} & $\dn$ & @{thm (rhs) code.simps(6)}\\
-  @{thm (lhs) code.simps(7)} & $\dn$ & @{thm (rhs) code.simps(7)}
-\end{tabular}
-\end{center}
-
-\begin{center}
-\begin{tabular}{lcl}
-  @{term areg} & $::=$ & @{term "AZERO"}\\
-               & $\mid$ & @{term "AONE bs"}\\
-               & $\mid$ & @{term "ACHAR bs c"}\\
-               & $\mid$ & @{term "AALT bs r1 r2"}\\
-               & $\mid$ & @{term "ASEQ bs r\<^sub>1 r\<^sub>2"}\\
-               & $\mid$ & @{term "ASTAR bs r"}
-\end{tabular}
-\end{center}
-
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) intern.simps(1)} & $\dn$ & @{thm (rhs) intern.simps(1)}\\
-  @{thm (lhs) intern.simps(2)} & $\dn$ & @{thm (rhs) intern.simps(2)}\\
-  @{thm (lhs) intern.simps(3)} & $\dn$ & @{thm (rhs) intern.simps(3)}\\
-  @{thm (lhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) intern.simps(6)} & $\dn$ & @{thm (rhs) intern.simps(6)}\\
-\end{tabular}
-\end{center}
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) erase.simps(1)} & $\dn$ & @{thm (rhs) erase.simps(1)}\\
-  @{thm (lhs) erase.simps(2)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(2)[of bs]}\\
-  @{thm (lhs) erase.simps(3)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(3)[of bs]}\\
-  @{thm (lhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) erase.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) erase.simps(6)[of bs]} & $\dn$ & @{thm (rhs) erase.simps(6)[of bs]}\\
-\end{tabular}
-\end{center}
-
-Some simple facts about erase
-
-\begin{lemma}\mbox{}\\
-@{thm erase_bder}\\
-@{thm erase_intern}
-\end{lemma}
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) bnullable.simps(1)} & $\dn$ & @{thm (rhs) bnullable.simps(1)}\\
-  @{thm (lhs) bnullable.simps(2)} & $\dn$ & @{thm (rhs) bnullable.simps(2)}\\
-  @{thm (lhs) bnullable.simps(3)} & $\dn$ & @{thm (rhs) bnullable.simps(3)}\\
-  @{thm (lhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bnullable.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bnullable.simps(6)} & $\dn$ & @{thm (rhs) bnullable.simps(6)}\medskip\\
-
-%  \end{tabular}
-%  \end{center}
-
-%  \begin{center}
-%  \begin{tabular}{lcl}
-
-  @{thm (lhs) bder.simps(1)} & $\dn$ & @{thm (rhs) bder.simps(1)}\\
-  @{thm (lhs) bder.simps(2)} & $\dn$ & @{thm (rhs) bder.simps(2)}\\
-  @{thm (lhs) bder.simps(3)} & $\dn$ & @{thm (rhs) bder.simps(3)}\\
-  @{thm (lhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(4)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bder.simps(5)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bder.simps(6)} & $\dn$ & @{thm (rhs) bder.simps(6)}
-  \end{tabular}
-  \end{center}
-
-
-\begin{center}
-  \begin{tabular}{lcl}
-  @{thm (lhs) bmkeps.simps(1)} & $\dn$ & @{thm (rhs) bmkeps.simps(1)}\\
-  @{thm (lhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(2)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bmkeps.simps(3)[of bs "r\<^sub>1" "r\<^sub>2"]}\\
-  @{thm (lhs) bmkeps.simps(4)} & $\dn$ & @{thm (rhs) bmkeps.simps(4)}\medskip\\
-\end{tabular}
-\end{center}
-
-
-@{thm [mode=IfThen] bder_retrieve}
-
-By induction on \<open>r\<close>
-
-\begin{theorem}[Main Lemma]\mbox{}\\
-@{thm [mode=IfThen] MAIN_decode}
-\end{theorem}
-
-\noindent
-Definition of the bitcoded lexer
-
-@{thm blexer_def}
-
-
-\begin{theorem}
-@{thm blexer_correctness}
-\end{theorem}
-
-\<close>
-\<close>*)
\ No newline at end of file